# Some factors of numbers of the form $b^n + / - 1$ found using ECM with new classes of curves

François Morain

► To cite this version:

François Morain. Some factors of numbers of the form $b^n + / - 1$ found using ECM with new classes of curves. 2022. hal-03437714v2

HAL Id: hal-03437714

https://inria.hal.science/hal-03437714v2

Preprint submitted on 31 Mar 2022

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Distributed under a Creative Commons Attribution 4.0 International License
SOME FACTORS OF NUMBERS OF THE FORM $b^n \pm 1$
FOUND USING ECM WITH NEW CLASSES OF CURVES

F. MORAIN

ABSTRACT. In this note, we list factors of numbers of the form $b^n \pm 1$ found using GMP-ECM and new classes of elliptic curves.

The elliptic curve method of factorization (ECM) [3] is one of the powerful algorithms able to find moderate size factors of large integers. The following tables list some unknown factors of composite numbers $b^n \pm 1$ for values of $13 \le b \le 10^4$, full tables of which are managed by Jonathan Crombie (continuing the work of Brent, Montgomery and te Riele, see http://myfactors.mooo.com/). This is an extension of the Cunningham project [1] for $2 \le b \le 12$. All the factors were found using GMP-ECM [6] (with the algorithms described in [5]).

More details on the computations and curves used are given in [4].

**Format of the tables:** pretty straightforward. Remember that $b, n+$ (resp. $b, n-$) designates $b^n + 1$ (resp. $b^n - 1$). The last column contains a time stamp. For instance, 210919 should be decoded as 21/09/19. Numbers are sorted by increasing date of discovery. More factorizations will be added as time goes by.

The last tables give factors found with the standard GMP-ECM using the traditional Suyama curves (labeled with $\mathbb{Z}/6\mathbb{Z}$) and the family Suyama-11 found by A. Kruppa [2].

REFERENCES

[1] J. Brillhart, D. H. Lehmer, J. L. Selfridge, B. Tuckerman, and S. S. Wagstaff, Jr. *Factorizations of $b^n \pm 1$, b = 2, 3, 5, 6, 7, 10, 11, 12 up to high powers*. Number 22 in Contemporary Mathematics. AMS, 2 edition, 1988.

[2] Alexander Kruppa. *Améliorations de la multiplication et de la factorisation d'entier*. PhD thesis, Univ. Nancy 1, 2010. Thèse de doctorat dirigée par Zimmermann, Paul.

[3] H. W. Lenstra, Jr. Factoring integers with elliptic curves. *Ann. of Math. (2)*, 126:649–673, 1987.

[4] François Morain. Modular curves over number fields and ECM. working paper or preprint, March 2022.

[5] P. Zimmermann and B. Dodson. 20 years of ECM. In Florian Hess, Sebastian Pauli, and Michael Pohst, editors, *Algorithmic Number Theory – ANTS-VII*, volume 4076 of *Lecture Notes in Comput. Sci.*, pages 525–542. Springer-Verlag, 2006.

[6] P. Zimmermann et al. GMP-ECM (elliptic curve method for integer factorization). https://gforge.inria.fr/projects/ecm/, 2010.

LIX - LABORATOIRE D'INFORMATIQUE DE L'ÉCOLE POLYTECHNIQUE, CNRS, INSTITUT POLYTECHNIQUE DE PARIS
and GRACE - INRIA SACLAY-ÎLE-DE-FRANCE

Email address: morain@lix.polytechnique.fr

<footnote>Date: March 31, 2022– update 1.</footnote>
<table>
  <thead>
    <tr>
      <th>N factor of (dd)</th>
      <th>p | N(dd)</th>
      <th>T, P(Z)</th>
      <th>when</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>91, 143 + (229dd)</td>
      <td>39097487455641709584561227838397 (32dd)</td>
      <td>Z/6Z × Z/6Z<br>Z<sup>2</sup> + 11</td>
      <td>210919</td>
    </tr>
    <tr>
      <td>⟨2·3⟩ × ⟨2·3<sup>2</sup>·13·41·79·13331·17401·1749887·21179897449⟩</td>
      <td></td>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <td>76, 143 - (226dd)</td>
      <td>1081673573275568956913253438874031 (34dd)</td>
      <td>Z/6Z × Z/6Z<br>Z<sup>2</sup> + 11</td>
      <td>210919</td>
    </tr>
    <tr>
      <td>⟨2⟩ × ⟨2<sup>2</sup>·3<sup>2</sup>·47·157·349·1123·297019·1104343·15836996297⟩</td>
      <td></td>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <td>89, 136 + (226dd)</td>
      <td>203670153440347727407773377 (27dd)</td>
      <td>Z/6Z × Z/6Z<br>Z<sup>2</sup> - 2</td>
      <td>210919</td>
    </tr>
    <tr>
      <td>⟨2⟩ × ⟨2·3·13·1039·6691·14011·3605783·3717299⟩</td>
      <td></td>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <td>91, 136 + (238dd)</td>
      <td>9909969840558744190100319969953 (31dd)</td>
      <td>Z/6Z × Z/6Z<br>Z<sup>2</sup> - 2</td>
      <td>210920</td>
    </tr>
    <tr>
      <td>⟨2⟩ × ⟨2·3·11·13·661·4793·554627·12295519·267299587⟩</td>
      <td></td>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <td>91, 136 + (238dd)</td>
      <td>889178099098009792925556006648298103113249 (41dd)</td>
      <td>Z/6Z × Z/6Z<br>Z<sup>2</sup> - 2</td>
      <td>210920</td>
    </tr>
    <tr>
      <td>⟨2⟩ × ⟨2·5<sup>2</sup>·7·29·2683·126473·266587·1652881·4049293·7234599079⟩</td>
      <td></td>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <td>85, 143 - (228dd)</td>
      <td>6314059662689865591498003869 (28dd)</td>
      <td>Z/6Z × Z/6Z<br>Z<sup>2</sup> + 11</td>
      <td>210920</td>
    </tr>
    <tr>
      <td>⟨2⟩ × ⟨2<sup>5</sup>·3<sup>2</sup>·7·13·2971·159157·12900367·19747601⟩</td>
      <td></td>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <td>83, 136 + (224dd)</td>
      <td>122131139150783253754177955960717201 (36dd)</td>
      <td>Z/6Z × Z/6Z<br>Z<sup>2</sup> - 2</td>
      <td>210920</td>
    </tr>
    <tr>
      <td>⟨2⟩ × ⟨2·3·5·3061·4091·41969·153889·1069463·23533167479⟩</td>
      <td></td>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <td>77, 148 + (246dd)</td>
      <td>6133384128714567850895531329681 (31dd)</td>
      <td>Z/4Z × Z/8Z<br>Z<sup>2</sup> - 37</td>
      <td>210920</td>
    </tr>
    <tr>
      <td>⟨2⟩ × ⟨2·13·83<sup>2</sup>·739·155747·7900859·18827913607⟩</td>
      <td></td>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <td>85, 136 + (244dd)</td>
      <td>4812510031276568181537132034273006315681 (40dd)</td>
      <td>Z/6Z × Z/6Z<br>Z<sup>2</sup> - 2</td>
      <td>210921</td>
    </tr>
    <tr>
      <td>⟨2⟩ × ⟨2<sup>4</sup>·7·41·83·193·1049·4483·33601·398357·5133307·101236763⟩</td>
      <td></td>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <td>90, 136 + (243dd)</td>
      <td>95029336145694838957824482284328662990992760116961 (50dd)</td>
      <td>Z/2Z × Z/10Z<br>Z<sup>2</sup> + 2</td>
      <td>211001</td>
    </tr>
    <tr>
      <td>⟨2⟩ × ⟨2·3<sup>3</sup>·5·7·17·23<sup>2</sup>·557·93169·191507·211093·3555857·19430611·19286145689⟩</td>
      <td></td>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <td>89, 136 + (226dd)</td>
      <td>1507236940222560786386041061745902423210609 (43dd)</td>
      <td>Z/2Z × Z/10Z<br>Z<sup>2</sup> + 2</td>
      <td>211002</td>
    </tr>
    <tr>
      <td>⟨2⟩ × ⟨2·3·5·89·641·2503·4099·65167·8986403·11806357·6207460211⟩</td>
      <td></td>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <td>86, 143 - (226dd)</td>
      <td>121802385644813164533936967364856084136235563 (45dd)</td>
      <td>Z/15Z<br>Z<sup>2</sup> + 11</td>
      <td>211002</td>
    </tr>
    <tr>
      <td>⟨3·5·101·331·383·1373·2417·114659·10160377·21286981·7706144731⟩</td>
      <td></td>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <td>91, 143 - (229dd)</td>
      <td>72384483510845642297891730646993695081321681523 (47dd)</td>
      <td>Z/15Z<br>Z<sup>2</sup> + 11</td>
      <td>211003</td>
    </tr>
    <tr>
      <td>⟨2·3·5·163·1627·2141·2617·28697·114913·10599097·10805321·4299485389⟩</td>
      <td></td>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <td>93, 136 + (224dd)</td>
      <td>144361306010767021849232771856031876990280881 (45dd)</td>
      <td>Z/2Z × Z/10Z<br>Z<sup>2</sup> + 2</td>
      <td>211004</td>
    </tr>
    <tr>
      <td>⟨2<sup>2</sup>⟩ × ⟨2<sup>2</sup>·3·11·37·691·3121·6571·94343·221707·237767·9395963·11159119⟩</td>
      <td></td>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <td>58, 148 + (245dd)</td>
      <td>1125068823066291949019180463529 (30dd)</td>
      <td>Z/4Z × Z/8Z<br>Z<sup>2</sup> - 37</td>
      <td>211014</td>
    </tr>
    <tr>
      <td>⟨2<sup>2</sup>⟩ × ⟨2<sup>3</sup>·3·132749·474119·13768663·1352376043⟩</td>
      <td></td>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <td>57, 148 + (243dd)</td>
      <td>169148129899316676754479151313 (30dd)</td>
      <td>Z/4Z × Z/8Z<br>Z<sup>2</sup> - 37</td>
      <td>211014</td>
    </tr>
    <tr>
      <td>⟨2⟩ × ⟨2·3·41·67·13425623·23712397·16118231531⟩</td>
      <td></td>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <td>57, 148 + (243dd)</td>
      <td>421009383140130353639417265637529 (33dd)</td>
      <td>Z/4Z × Z/8Z<br>Z<sup>2</sup> - 37</td>
      <td>211014</td>
    </tr>
    <tr>
      <td>⟨2<sup>2</sup>⟩ × ⟨2<sup>5</sup>·31·71<sup>2</sup>·397·568049·3578951·26077824329⟩</td>
      <td></td>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <td>59, 148 + (246dd)</td>
      <td>26642397493750326276664956281 (29dd)</td>
      <td>Z/4Z × Z/8Z<br>Z<sup>2</sup> - 37</td>
      <td>211014</td>
    </tr>
    <tr>
      <td>⟨2⟩ × ⟨2·3·11·139·631·17351·5071639·26150642897⟩</td>
      <td></td>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <td>63, 148 + (239dd)</td>
      <td>24427916239531829701088033849 (29dd)</td>
      <td>Z/4Z × Z/8Z<br>Z<sup>2</sup> - 37</td>
      <td>211014</td>
    </tr>
    <tr>
      <td>⟨2<sup>2</sup>⟩ × ⟨2<sup>3</sup>·359·7757·92233·359377·8270115787⟩</td>
      <td></td>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <td>63, 148 + (239dd)</td>
      <td>126775587192783617996871593158271705801 (39dd)</td>
      <td>Z/4Z × Z/8Z<br>Z<sup>2</sup> - 37</td>
      <td>211015</td>
    </tr>
    <tr>
      <td>⟨2<sup>3</sup>⟩ × ⟨2<sup>3</sup>·3<sup>3</sup>·11·1193·66797·411799·505213·11881027·33860083⟩</td>
      <td></td>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <td>62, 148 + (231dd)</td>
      <td>1744893892759353490478092942757189489 (36dd)</td>
      <td>Z/4Z × Z/8Z<br>Z<sup>2</sup> - 37</td>
      <td>211015</td>
    </tr>
    <tr>
      <td>⟨2⟩ × ⟨2·3·1699·20921·23971·100799·2135723·79272883927⟩</td>
      <td></td>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <td>75, 148 + (239dd)</td>
      <td>287256660129937632149933799072015793 (36dd)</td>
      <td>Z/4Z × Z/8Z<br>Z<sup>2</sup> - 37</td>
      <td>211015</td>
    </tr>
    <tr>
      <td>⟨2<sup>2</sup>⟩ × ⟨2<sup>9</sup>·3·5·7<sup>4</sup>·29·14281·3271661·4630721·620702087⟩</td>
      <td></td>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <td>72, 143 + (219dd)</td>
      <td>257046566603437226512911589147 (30dd)</td>
      <td>Z/6Z × Z/6Z<br>Z<sup>2</sup> + 11</td>
      <td>211016</td>
    </tr>
    <tr>
      <td>⟨2·3⟩ × ⟨2<sup>4</sup>·3<sup>3</sup>·5·7<sup>2</sup>·1471·1999·40892587·3366205867⟩</td>
      <td></td>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <td>95, 143 + (223dd)</td>
      <td>1083464823584081602004225920974229961236997 (43dd)</td>
      <td>Z/6Z × Z/6Z<br>Z<sup>2</sup> + 11</td>
      <td>211016</td>
    </tr>
    <tr>
      <td>⟨2⟩ × ⟨2<sup>4</sup>·3·5<sup>3</sup>·163·179·251·5591·14011·217337·256639·778027·3626683⟩</td>
      <td></td>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <td>63, 148 + (239dd)</td>
      <td>2708217308807273429767899154174901407393 (40dd)</td>
      <td>Z/4Z × Z/8Z<br>Z<sup>2</sup> - 37</td>
      <td>211017</td>
    </tr>
    <tr>
      <td>⟨2⟩ × ⟨2·5·19·677927·3787247·8691433·9805921·32569686239⟩</td>
      <td></td>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <td>59, 148 + (246dd)</td>
      <td>6153600891183451358288633718234351691755490502156977 (52dd)</td>
      <td>Z/4Z × Z/8Z<br>Z<sup>2</sup> - 37</td>
      <td>211019</td>
    </tr>
    <tr>
      <td>⟨2<sup>2</sup>⟩ × ⟨2<sup>4</sup>·3<sup>4</sup>·31·383·659·12413·44087·176261·269231·4538333·5268647·244317397⟩</td>
      <td></td>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <td>80, 148 + (249dd)</td>
      <td>528008828433104496135816963159066412667528154401 (48dd)</td>
      <td>Z/4Z × Z/8Z<br>Z<sup>2</sup> - 37</td>
      <td>211019</td>
    </tr>
    <tr>
      <td>⟨2<sup>2</sup>⟩ × ⟨2<sup>4</sup>·3·1951·3121·9923·23911·196919·1851271·31834379·164019009337⟩</td>
      <td></td>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <td>68, 148 + (226dd)</td>
      <td>6293335603138665673753869409668883438923628000153 (49dd)</td>
      <td>Z/4Z × Z/8Z<br>Z<sup>2</sup> - 37</td>
      <td>211023</td>
    </tr>
    <tr>
      <td>⟨2<sup>2</sup>·3⟩ × ⟨2<sup>4</sup>·3·7·11·101·683<sup>2</sup>·1277·214309·509783·3968491·28613441·95052329⟩</td>
      <td></td>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <td>58, 148 + (245dd)</td>
      <td>195506542210979176522718797760192511806277913 (45dd)</td>
      <td>Z/4Z × Z/8Z<br>Z<sup>2</sup> - 37</td>
      <td>211026</td>
    </tr>
    <tr>
      <td>⟨2⟩ × ⟨2·79·367·1487·20357·55639·268883·377711·1576543·6251371211⟩</td>
      <td></td>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <td>59, 148 + (246dd)</td>
      <td>32549598165881659323145053260648245905788521 (44dd)</td>
      <td>Z/4Z × Z/8Z<br>Z<sup>2</sup> - 37</td>
      <td>211027</td>
    </tr>
    <tr>
      <td>⟨2⟩ × ⟨2·17·461·1013·6709·13241·21839·1434023·1765697·208661692721⟩</td>
      <td></td>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <td>95, 145 - (222dd)</td>
      <td>200665557019668388820419786457045587248891341 (45dd)</td>
      <td>Z/4Z × Z/8Z<br>Z<sup>2</sup> - 29</td>
      <td>211030</td>
    </tr>
    <tr>
      <td>⟨2⟩ × ⟨2·3<sup>2</sup>·7·3923·31277·160621·477209·4281349·18694493·1057848859⟩</td>
      <td></td>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <td>88, 143 + (215dd)</td>
      <td>6358121600282021249167413373357836215012153 (43dd)</td>
      <td>Z/6Z × Z/6Z<br>Z<sup>2</sup> + 11</td>
      <td>211103</td>
    </tr>
    <tr>
      <td>⟨2⟩ × ⟨2<sup>2</sup>·3·137·499·569·1429·4703·22447·319733·1806031·78181865953⟩</td>
      <td></td>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <td>74, 145 + (210dd)</td>
      <td>3242269523406605838609691912552260883503075877086401 (52dd)</td>
      <td>Z/4Z × Z/8Z<br>Z<sup>2</sup> - 5</td>
      <td>211103</td>
    </tr>
    <tr>
      <td>⟨2<sup>2</sup>⟩ × ⟨2<sup>3</sup>·11·13·53·839·1427·32647·658663·792277·1532647·8783009·48689154383⟩</td>
      <td></td>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <td>80, 136 + (208dd)</td>
      <td>14282340552001320145244125431697889 (35dd)</td>
      <td>Z/6Z × Z/6Z<br>Z<sup>2</sup> - 2</td>
      <td>211104</td>
    </tr>
    <tr>
      <td>⟨2⟩ × ⟨2·3<sup>2</sup>·4243·1238693·1258783·5029561·11922826373⟩</td>
      <td></td>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <td>82, 145 + (211dd)</td>
      <td>8674789605346661424832133115073007844671 (40dd)</td>
      <td>Z/4Z × Z/8Z<br>Z<sup>2</sup> - 5</td>
      <td>211104</td>
    </tr>
    <tr>
      <td>⟨2⟩ × ⟨2<sup>5</sup>·7<sup>2</sup>·31·173·503·2143·3083·5270231·38727967·760426193⟩</td>
      <td></td>
      <td></td>
      <td></td>
    </tr>
  </tbody>
</table>
<table><thead><tr><th>N factor of (dd)</th><th>p | N(dd)</th><th>T, P(Z)</th><th>when</th></tr></thead><tbody><tr><td>69, 145 + (206dd)</td><td>286508193459105224545095761 (27dd)<br/>(2<sup>2</sup>) × (2<sup>5</sup> · 7 · 29 · 71<sup>2</sup> · 44273 · 197273 · 250442333)</td><td>Z/4Z × Z/8Z<br/>Z<sup>2</sup> - 5</td><td>211104</td></tr><tr><td>82, 145 + (171dd)</td><td>684748065015589423265391174612848221496545022161 (48dd)<br/>(2<sup>2</sup>) × (2<sup>3</sup> · 3<sup>2</sup> · 13 · 37 · 47 · 67 · 84551 · 885977 · 1071803 · 2396239 · 16453693 · 495873503)</td><td>Z/4Z × Z/8Z<br/>Z<sup>2</sup> - 5</td><td>211105</td></tr><tr><td>97, 143 + (205dd)</td><td>5029284168421691009474960293330997501 (37dd)<br/>(2<sup>2</sup>) × (2<sup>3</sup> · 3 · 5 · 29 · 1381 · 34519 · 36847 · 44651 · 49523 · 93019668149)</td><td>Z/6Z × Z/6Z<br/>Z<sup>2</sup> + 11</td><td>211105</td></tr><tr><td>908, 67 - (190dd)</td><td>3149249925200951868613704379189470025083148513889 (48dd)<br/>(2) × (2 · 3 · 7<sup>2</sup> · 17 · 47 · 127 · 157 · 503 · 54277 · 683759 · 11269243 · 16866691 · 9474758309)</td><td>Z/2Z × Z/12Z<br/>Z<sup>3</sup> + Z<sup>2</sup> - 22Z + 5</td><td>211105</td></tr><tr><td>832, 67 - (191dd)</td><td>5152060195650213711128948395000809254091832789 (46dd)<br/>(2<sup>2</sup>) × (2<sup>3</sup> · 3 · 103 · 311 · 1123 · 9277 · 585283 · 1013029 · 2068117 · 2756731 · 47573761)</td><td>Z/2Z × Z/12Z<br/>Z<sup>3</sup> + Z<sup>2</sup> - 22Z + 5</td><td>211105</td></tr><tr><td>85, 143 + (211dd)</td><td>125672102977571569625797213226021008844808941 (46dd)<br/>(2) × (2 · 3 · 14731 · 124213 · 314719 · 741991 · 990137 · 19902737 · 12437373079)</td><td>Z/6Z × Z/6Z<br/>Z<sup>2</sup> + 11</td><td>211106</td></tr><tr><td>80, 136 + (173dd)</td><td>1779586756730767305188268537007290668449 (40dd)<br/>(2) × (2 · 5<sup>2</sup> · 29 · 89 · 14387 · 11752711 · 12929507 · 32430857 · 97248401)</td><td>Z/6Z × Z/6Z<br/>Z<sup>2</sup> - 2</td><td>211106</td></tr><tr><td>78, 145 + (207dd)</td><td>810726926248106462415561622241332161181 (39dd)<br/>(2<sup>2</sup>) × (2<sup>5</sup> · 7 · 19 · 41 · 59 · 257 · 269 · 607 · 15401 · 26227 · 24356611 · 47685751)</td><td>Z/4Z × Z/8Z<br/>Z<sup>2</sup> - 5</td><td>211106</td></tr><tr><td>77, 145 + (204dd)</td><td>5351129477105280700750350283769394807311831 (43dd)<br/>(2) × (2<sup>3</sup> · 3<sup>3</sup> · 211 · 269 · 853 · 12203 · 75797 · 3532457 · 4405043 · 17775927973)</td><td>Z/4Z × Z/8Z<br/>Z<sup>2</sup> - 5</td><td>211106</td></tr><tr><td>77, 145 + (161dd)</td><td>15284105262481307713678612469222143741121 (41dd)<br/>(2) × (2 · 3 · 17 · 233 · 211187 · 644687 · 8579891 · 32402311 · 8495326627)</td><td>Z/4Z × Z/8Z<br/>Z<sup>2</sup> - 5</td><td>211107</td></tr><tr><td>61, 148 + (206dd)</td><td>5392083635732772033735788905270233582687837357169 (49dd)<br/>(2<sup>4</sup>) × (2<sup>4</sup> · 3<sup>2</sup> · 5 · 7 · 17 · 23<sup>2</sup> · 317 · 359 · 18133 · 6378881 · 12983833 · 35228899 · 1234899581)</td><td>Z/4Z × Z/8Z<br/>Z<sup>2</sup> - 37</td><td>211107</td></tr><tr><td>78, 143 - (203dd)</td><td>58906988500210848827148393647 (29dd)<br/>(2) × (2<sup>4</sup> · 3<sup>2</sup> · 7 · 83 · 241 · 8053 · 13999 · 179471 · 72199163)</td><td>Z/6Z × Z/6Z<br/>Z<sup>2</sup> + 11</td><td>211109</td></tr><tr><td>69, 145 + (180dd)</td><td>51642735295054165114200696495014578655491621 (44dd)<br/>(2) × (2 · 3 · 7 · 11 · 167 · 1901 · 13903 · 100313 · 279007 · 286721 · 39254783 · 40198121)</td><td>Z/4Z × Z/8Z<br/>Z<sup>2</sup> - 5</td><td>211109</td></tr><tr><td>71, 145 + (200dd)</td><td>3081838728839138646824667134834215417088225761 (46dd)<br/>(2 · 3 · 5) × (2 · 3<sup>3</sup> · 5 · 43 · 8831 · 27691 · 5737181 · 9128929 · 42885001 · 16109575787)</td><td>Z/4Z × Z/8Z<br/>Z<sup>2</sup> - 5</td><td>211110</td></tr><tr><td>74, 145 - (198dd)</td><td>115699409434579757150069122314705902015821 (42dd)<br/>(2<sup>2</sup> · 5) × (2<sup>3</sup> · 3<sup>2</sup> · 5 · 347 · 367 · 2927 · 18059 · 1288657 · 3614153 · 512557762621)</td><td>Z/4Z × Z/8Z<br/>Z<sup>2</sup> - 5</td><td>211110</td></tr><tr><td>74, 145 - (116dd)</td><td>672757891558517597623408574883629411681 (39dd)<br/>(2) × (2 · 19 · 61 · 256603 · 280541 · 4038467 · 11303779 · 44158784113)</td><td>Z/4Z × Z/8Z<br/>Z<sup>2</sup> - 5</td><td>211110</td></tr><tr><td>74, 145 - (157dd)</td><td>82680336163400607564958279513497353433701 (41dd)<br/>(2) × (2 · 3<sup>3</sup> · 31 · 179 · 1831 · 95701 · 121403 · 153521 · 11527619 · 3664562303)</td><td>Z/4Z × Z/8Z<br/>Z<sup>2</sup> - 5</td><td>211110</td></tr><tr><td>737, 67 + (184dd)</td><td>870917417466838788698821597667901172952093040299 (48dd)<br/>(2) × (2<sup>3</sup> · 3 · 5 · 19 · 907 · 5413 · 8807 · 22973 · 553103 · 4764239 · 6769901 · 10778026289)</td><td>Z/2Z × Z/12Z<br/>Z<sup>3</sup> + Z<sup>2</sup> - 22Z + 5</td><td>211111</td></tr><tr><td>682, 67 + (185dd)</td><td>3960431214365287888129140722498272391957628371 (46dd)<br/>(2) × (2<sup>3</sup> · 3<sup>2</sup> · 13 · 67 · 5441 · 11353 · 19319 · 59581 · 319399 · 13319101 · 104393045839)</td><td>Z/2Z × Z/12Z<br/>Z<sup>3</sup> + Z<sup>2</sup> - 22Z + 5</td><td>211112</td></tr><tr><td>68, 145 + (201dd)</td><td>759522532722573631464900168860811795924541 (42dd)<br/>(2<sup>2</sup>) × (2<sup>6</sup> · 3<sup>3</sup> · 11 · 83 · 173 · 852833 · 952507 · 1032509 · 5399377 · 153620927)</td><td>Z/4Z × Z/8Z<br/>Z<sup>2</sup> - 5</td><td>211112</td></tr><tr><td>79, 136 + (212dd)</td><td>1676257114889837463366010414743038265629041 (43dd)<br/>(2 · 5) × (2 · 3<sup>2</sup> · 5 · 43<sup>2</sup> · 97 · 317 · 419 · 1493 · 4051 · 152617 · 1069429 · 79202570021)</td><td>Z/2Z × Z/10Z<br/>Z<sup>2</sup> + 2</td><td>211113</td></tr><tr><td>79, 143 + (201dd)</td><td>617165874719529966578857601636808679159 (39dd)<br/>(2 · 3) × (2<sup>2</sup> · 3 · 5 · 12959 · 3388459 · 23405677 · 29667019 · 56225142067)</td><td>Z/6Z × Z/6Z<br/>Z<sup>2</sup> + 11</td><td>211113</td></tr><tr><td>564, 67 - (182dd)</td><td>12006347502653555973091239781599251958537240793 (46dd)<br/>(2) × (2<sup>2</sup> · 3<sup>2</sup> · 5 · 293 · 225089 · 459509 · 1811561 · 4338643 · 8046043 · 1740216437)</td><td>Z/2Z × Z/12Z<br/>Z<sup>3</sup> + Z<sup>2</sup> - 22Z + 5</td><td>211114</td></tr><tr><td>69, 143 + (195dd)</td><td>46032704990894163010799652245639 (32dd)<br/>(2) × (2<sup>4</sup> · 3 · 5 · 23 · 797 · 21673 · 990589 · 13396063 · 18190699)</td><td>Z/6Z × Z/6Z<br/>Z<sup>2</sup> + 11</td><td>211114</td></tr><tr><td>97, 145 + (195dd)</td><td>20475011708627365490421549472180159901 (38dd)<br/>(2<sup>2</sup>) × (2<sup>4</sup> · 5<sup>3</sup> · 7 · 11<sup>2</sup> · 2731 · 4831 · 10691 · 15767 · 896047 · 1516330381)</td><td>Z/4Z × Z/8Z<br/>Z<sup>2</sup> - 5</td><td>211115</td></tr><tr><td>75, 145 - (195dd)</td><td>145820922799205411577526521075547603991 (39dd)<br/>(2) × (2<sup>5</sup> · 3<sup>2</sup> · 5<sup>3</sup> · 71317 · 671257 · 1601317 · 12411877 · 2128583419)</td><td>Z/4Z × Z/8Z<br/>Z<sup>2</sup> - 5</td><td>211115</td></tr><tr><td>76, 136 + (193dd)</td><td>230638965208492675830062875068689 (33dd)<br/>(2<sup>2</sup>) × (2<sup>2</sup> · 3 · 13 · 43 · 3037 · 3134347 · 28164953 · 32061123557)</td><td>Z/6Z × Z/6Z<br/>Z<sup>2</sup> - 2</td><td>211115</td></tr><tr><td>87, 145 + (194dd)</td><td>59493967220771242233729202831 (29dd)<br/>(2) × (2<sup>4</sup> · 239 · 3967 · 22481 · 691697 · 126104717161)</td><td>Z/4Z × Z/8Z<br/>Z<sup>2</sup> - 5</td><td>211115</td></tr><tr><td>98, 145 - (161dd)</td><td>106565478545504059784085620836190321 (36dd)<br/>(2<sup>3</sup>) × (2<sup>3</sup> · 53 · 521 · 827 · 34679 · 43661 · 6122341 · 7865742119)</td><td>Z/4Z × Z/8Z<br/>Z<sup>2</sup> - 5</td><td>211116</td></tr><tr><td>97, 145 + (158dd)</td><td>20325839679108068942542957640818669021 (38dd)<br/>(2<sup>2</sup>) × (2<sup>3</sup> · 43 · 101 · 1217 · 95723 · 271981 · 41395811 · 111508210703)</td><td>Z/4Z × Z/8Z<br/>Z<sup>2</sup> - 5</td><td>211116</td></tr><tr><td>89, 145 - (192dd)</td><td>24305667473968387305296796282225547921 (39dd)<br/>(2) × (2<sup>3</sup> · 11 · 13 · 18229 · 232753 · 6409253 · 14979817 · 260783119673)</td><td>Z/4Z × Z/8Z<br/>Z<sup>2</sup> - 5</td><td>211116</td></tr><tr><td>76, 136 + (131dd)</td><td>438641314708689994945803094414364017 (36dd)<br/>(2 · 3) × (2 · 3 · 11 · 17<sup>2</sup> · 211 · 9437 · 7419649 · 12814147 · 20245431293)</td><td>Z/6Z × Z/6Z<br/>Z<sup>2</sup> - 2</td><td>211116</td></tr><tr><td>76, 136 + (161dd)</td><td>156680649601795675183141661297 (30dd)<br/>(2) × (2<sup>3</sup> · 3 · 5 · 11 · 191 · 1481 · 36871 · 6726761 · 845925461)</td><td>Z/6Z × Z/6Z<br/>Z<sup>2</sup> - 2</td><td>211116</td></tr><tr><td>98, 145 - (192dd)</td><td>4538131062026968680387147363911 (31dd)<br/>(2) × (2<sup>7</sup> · 29167 · 2240369 · 6780707 · 40008362269)</td><td>Z/4Z × Z/8Z<br/>Z<sup>2</sup> - 5</td><td>211116</td></tr><tr><td>88, 143 - (205dd)</td><td>176631419599003306284596237977817343802484627 (45dd)<br/>(2<sup>2</sup> · 3<sup>2</sup> · 5 · 79 · 1789 · 57191 · 59093 · 212437 · 2217011 · 3417079 · 1276560101)</td><td>Z/15Z<br/>Z<sup>2</sup> + 11</td><td>211116</td></tr></tbody></table>
<table><thead><tr><th>N factor of (dd)</th><th>p | N(dd)</th><th>T, P(Z)</th><th>when</th></tr></thead><tbody><tr><td>93, 143 + (185dd)</td><td>1107143747609638086483933799 (28dd)<br/>(2·3) × (2<sup>4</sup>·3<sup>2</sup>·5·11·71·193·2569607·661675745569)</td><td>Z/6Z × Z/6Z<br/>Z<sup>2</sup> + 11</td><td>211117</td></tr><tr><td>54, 148 + (142dd)</td><td>47745297013416763673913373201 (29dd)<br/>(2<sup>3</sup>) × (2<sup>5</sup>·7<sup>2</sup>·13·19·1021·3527·372523·11487176849)</td><td>Z/4Z × Z/8Z<br/>Z<sup>2</sup> - 37</td><td>211117</td></tr><tr><td>76, 136 + (96dd)</td><td>748413049020504684792641386896154057664353 (42dd)<br/>(2<sup>2</sup>) × (2<sup>2</sup>·5<sup>2</sup>·151·44789·283837·289889·9813829·11900699·28788653)</td><td>Z/6Z × Z/6Z<br/>Z<sup>2</sup> - 2</td><td>211117</td></tr><tr><td>89, 145 - (154dd)</td><td>267697400207700352494793748295552781 (36dd)<br/>(2<sup>2</sup>) × (2<sup>3</sup>·3·5·61·137·313·1433·195869·2151203·39234884243)</td><td>Z/4Z × Z/8Z<br/>Z<sup>2</sup> - 5</td><td>211117</td></tr><tr><td>71, 143 + (185dd)</td><td>6147619877984764806596458300991243997791 (40dd)<br/>(2<sup>2</sup>) × (2<sup>3</sup>·3·1627·6481·5016863·5558669·31165907·13975049939)</td><td>Z/6Z × Z/6Z<br/>Z<sup>2</sup> + 11</td><td>211117</td></tr><tr><td>87, 143 - (186dd)</td><td>869739041700065774535952087462183871 (36dd)<br/>(2<sup>2</sup>) × (2<sup>2</sup>·3·73<sup>2</sup>·2557·46261·206081·1921849·145154011963)</td><td>Z/6Z × Z/6Z<br/>Z<sup>2</sup> + 11</td><td>211117</td></tr><tr><td>54, 148 + (181dd)</td><td>1873736382664056942125771790331768514929 (40dd)<br/>(2<sup>2</sup>) × (2<sup>3</sup>·510619·1175107·19322837·25780987·195890645359)</td><td>Z/4Z × Z/8Z<br/>Z<sup>2</sup> - 37</td><td>211117</td></tr><tr><td>54, 148 + (113dd)</td><td>79647427812727331528378671705173747075361 (41dd)<br/>(2<sup>2</sup>) × (2<sup>4</sup>·3<sup>2</sup>·127·137·179·2857·6761·6365773·7789627·46353546331)</td><td>Z/4Z × Z/8Z<br/>Z<sup>2</sup> - 37</td><td>211119</td></tr><tr><td>84, 145 - (177dd)</td><td>1280935678670178982674695168311861 (34dd)<br/>(2<sup>2</sup>·3) × (2<sup>6</sup>·3<sup>3</sup>·17·179·2389·201769·1102979·38182340899)</td><td>Z/4Z × Z/8Z<br/>Z<sup>2</sup> - 5</td><td>211119</td></tr><tr><td>87, 143 - (150dd)</td><td>136899949256066917953836347405696564733147 (42dd)<br/>(2<sup>2</sup>) × (2<sup>3</sup>·3·23·29<sup>2</sup>·211·38891·142867·883763·19283413·7379977013)</td><td>Z/6Z × Z/6Z<br/>Z<sup>2</sup> + 11</td><td>211119</td></tr><tr><td>85, 145 - (136dd)</td><td>186647494681298441991785036724963841 (36dd)<br/>(2<sup>3</sup>) × (2<sup>3</sup>·13·293·571·10939·2447527·22052477·2271080971)</td><td>Z/4Z × Z/8Z<br/>Z<sup>2</sup> - 5</td><td>211120</td></tr><tr><td>85, 145 - (174dd)</td><td>20183280853192927742502546329975236601 (39dd)<br/>(2<sup>2</sup>) × (2<sup>3</sup>·13·53·421·7187·44171·311453·26625967·66076597283)</td><td>Z/4Z × Z/8Z<br/>Z<sup>2</sup> - 5</td><td>211120</td></tr><tr><td>80, 143 - (184dd)</td><td>6712531994782750141515031166298684148691 (40dd)<br/>(2<sup>2</sup>) × (2<sup>7</sup>·3<sup>3</sup>·113<sup>2</sup>·659·5641·10604969·26495507·72811939633)</td><td>Z/6Z × Z/6Z<br/>Z<sup>2</sup> + 11</td><td>211121</td></tr><tr><td>67, 145 + (185dd)</td><td>7248137329293656381576244670183065997056221 (43dd)<br/>(2<sup>2</sup>·5) × (2<sup>5</sup>·3<sup>2</sup>·5<sup>2</sup>·1531·5867·14731·40819·5329151·10308269·169642351)</td><td>Z/4Z × Z/8Z<br/>Z<sup>2</sup> - 5</td><td>211202</td></tr><tr><td>80, 143 - (145dd)</td><td>795054288520025014495743423224943946219831 (42dd)<br/>(2<sup>2</sup>) × (2<sup>3</sup>·3·31·20743·81533·188701·3968897·8522791·197980759619)</td><td>Z/2Z × Z/10Z<br/>Z<sup>2</sup> + 11</td><td>211203</td></tr><tr><td>66, 145 + (171dd)</td><td>8412817727511578850575210670325289161 (37dd)<br/>(2<sup>2</sup>) × (2<sup>3</sup>·3·7·6131·4273873·8884949·17883043·3006927227)</td><td>Z/4Z × Z/8Z<br/>Z<sup>2</sup> - 5</td><td>211204</td></tr><tr><td>99, 145 + (162dd)</td><td>4062412002827262841252226180328415331 (37dd)<br/>(2<sup>2</sup>) × (2<sup>5</sup>·3<sup>3</sup>·5<sup>2</sup>·173·367·29221·937033·5866457·9220664083)</td><td>Z/4Z × Z/8Z<br/>Z<sup>2</sup> - 5</td><td>211204</td></tr><tr><td>90, 145 + (170dd)</td><td>2009042889795473718562376631715120443361 (40dd)<br/>(2<sup>2</sup>) × (2<sup>3</sup>·13·139<sup>2</sup>·937·5393·9467·295777·1447913·97603863479)</td><td>Z/4Z × Z/8Z<br/>Z<sup>2</sup> - 5</td><td>211204</td></tr><tr><td>69, 143 - (163dd)</td><td>2075379844525804919871899022208031 (34dd)<br/>(2<sup>2</sup>) × (2<sup>3</sup>·5·1033·205619·390877·7049473·1858484281)</td><td>Z/6Z × Z/6Z<br/>Z<sup>2</sup> + 11</td><td>211205</td></tr><tr><td>90, 133 - (159dd)</td><td>1754145264557559798796607809202906289 (36dd)<br/>(2<sup>2</sup>) × (2<sup>2</sup>·13·139·487·12391·16945133·22387667·2650319803)</td><td>Z/6Z × Z/6Z<br/>Z<sup>2</sup> + 7</td><td>211205</td></tr><tr><td>67, 143 + (156dd)</td><td>694434947644982935847278082066102027988051029 (46dd)<br/>(2<sup>2</sup>) × (2<sup>3</sup>·3·1607·13147·78713·125497·284173·54698899·96717919·184441967)</td><td>Z/6Z × Z/6Z<br/>Z<sup>2</sup> + 11</td><td>211207</td></tr><tr><td>96, 145 - (142dd)</td><td>859157706616161686886486187181029171 (36dd)<br/>(2<sup>2</sup>) × (2<sup>3</sup>·3<sup>2</sup>·12619·708839·19993777·27569123·2420193299)</td><td>Z/4Z × Z/8Z<br/>Z<sup>2</sup> - 5</td><td>211208</td></tr><tr><td>97, 145 - (211dd)</td><td>17794182375018474055246335530679440628632195141 (47dd)<br/>(2<sup>2</sup>) × (2<sup>3</sup>·11·29·107·3769·4127·84713·105389·1981159·10157729·46636041451)</td><td>Z/4Z × Z/8Z<br/>Z<sup>2</sup> - 5</td><td>211213</td></tr><tr><td>61, 148 + (157dd)</td><td>38086824690111916319348286749664525448870608289 (47dd)<br/>(2<sup>2</sup>) × (2<sup>4</sup>·3<sup>2</sup>·997·2917·79823·202907·2167183·4802927·7505633·17968289)</td><td>Z/4Z × Z/8Z<br/>Z<sup>2</sup> - 37</td><td>211215</td></tr><tr><td>85, 143 + (165dd)</td><td>595429189435570985758361988110442572149227287 (45dd)<br/>(2<sup>2</sup>) × (2<sup>4</sup>·5·13·31·431·683·7883·17921·137369·177979·232081·39134182489)</td><td>Z/4Z × Z/8Z<br/>Z<sup>2</sup> - 13</td><td>211215</td></tr><tr><td>90, 145 + (131dd)</td><td>564048033596387231766876553229115117359161 (42dd)<br/>(2<sup>3</sup>) × (2<sup>4</sup>·3<sup>2</sup>·5·199·241781·2572373·4237813·10764197·17344468319)</td><td>Z/4Z × Z/8Z<br/>Z<sup>2</sup> - 5</td><td>211220</td></tr><tr><td>88, 145 + (199dd)</td><td>3305395902472416582389095389547828872322740179581 (49dd)<br/>(2<sup>2</sup>) × (2<sup>4</sup>·13<sup>2</sup>·29·613·1433·7127·25423·148921·2692159·24762499·6669103631)</td><td>Z/4Z × Z/8Z<br/>Z<sup>2</sup> - 29</td><td>220103</td></tr><tr><td>69, 145 - (196dd)</td><td>43973740306053298612759811200943577004074844768891 (50dd)<br/>(2<sup>2</sup>) × (2<sup>3</sup>·5·13·1381·834469·1456837·3504673·29722321·37912759·6377193661)</td><td>Z/4Z × Z/8Z<br/>Z<sup>2</sup> - 29</td><td>220103</td></tr><tr><td>96, 143 - (192dd)</td><td>1838435702793007470702888346700618324463818611 (46dd)<br/>(2<sup>2</sup>·5·13·179·811·12541·124919·1160519·6762571·30906133·128183399)</td><td>Z/14Z<br/>Z<sup>2</sup> + 11</td><td>220104</td></tr><tr><td>71, 143 - (173dd)</td><td>197679349184370896695764017247401612048842003 (45dd)<br/>(2<sup>2</sup>) × (2<sup>3</sup>·5·7·13·101<sup>2</sup>·293·1831·1489937·2513249·17506441·11213111743)</td><td>Z/15Z<br/>Z<sup>2</sup> + 11</td><td>220107</td></tr><tr><td>80, 140 + (145dd)</td><td>6919402781603403339139011273606732443068241 (43dd)<br/>(2<sup>2</sup>) × (2<sup>2</sup>·3<sup>3</sup>·13·19·23·31·43·353·631·1789·42841·594917·4403857·94579351)</td><td>Z/6Z × Z/6Z<br/>Z<sup>2</sup> + 7</td><td>220109</td></tr><tr><td>517, 67 - (180dd)</td><td>43673375920790464063197381024563082702565341226931 (50dd)<br/>(2<sup>2</sup>) × (2<sup>4</sup>·3<sup>5</sup>·11·17·31<sup>2</sup>·47·40151·500389·717341·761489·1183837·51181421299)</td><td>Z/2Z × Z/12Z<br/>Z<sup>3</sup> + Z<sup>2</sup> - 22Z + 5</td><td>220112</td></tr><tr><td>69, 145 + (136dd)</td><td>2312935912505799788408225250017130492549337428188531 (52dd)<br/>(2<sup>2</sup>) × (2<sup>3</sup>·11·13·43·66499·236681·351023·1047667·3274151·18302677·135555908207)</td><td>Z/4Z × Z/8Z<br/>Z<sup>2</sup> - 29</td><td>220115</td></tr><tr><td>93, 143 + (157dd)</td><td>169596120610870220111027662114795982337359989 (45dd)<br/>(2<sup>2</sup>) × (2<sup>5</sup>·7·20507·38543·538397·676979·6587479·7787239·81970891)</td><td>Z/4Z × Z/8Z<br/>Z<sup>2</sup> - 13</td><td>220118</td></tr><tr><td>97, 143 + (168dd)</td><td>845758389579765061876120422286830800293217720347603 (51dd)<br/>(2<sup>3</sup>·5·47·2281·3323·9643·20507·199523·617473·697373·8843827·58519211)</td><td>Z/15Z<br/>Z<sup>2</sup> - 13</td><td>220131</td></tr></tbody></table>
<table><thead><tr><th>N factor of (dd)</th><th>p | N(dd)</th><th>T, P(Z)</th><th>when</th></tr></thead><tbody><tr><td>74, 140 + (145dd)</td><td>54246844404658638528674908309645373687448082321 (47dd)</td><td>Z/15Z</td><td>220202</td></tr><tr><td>(3 · 5 · 19 · 2341 · 5827 · 1946183 · 2736637 · 3982073 · 9183641 · 71640433219)</td><td></td><td>Z<sup>2</sup> + 7</td><td></td></tr><tr><td>93, 143 + (113dd)</td><td>153122642255322360024123283032581744979354437 (45dd)</td><td>Z/2Z × Z/10Z</td><td>220203</td></tr><tr><td>(2) × (2<sup>3</sup> · 16381 · 344171 · 424169 · 786629 · 1944853 · 8991293 · 290928949)</td><td></td><td>Z<sup>2</sup> − 13</td><td></td></tr></tbody></table>

<table><thead><tr><th>N factor of (dd)</th><th>p | N(dd)</th><th>T</th><th>when</th></tr></thead><tbody><tr><td>77, 143 - (176dd)</td><td>137201052117407925881545888920089773400305732245589 (51dd)</td><td>Z/6Z</td><td>220120</td></tr><tr><td>(2<sup>6</sup> · 3<sup>2</sup> · 5 · 37 · 181 · 271 · 1373 · 1260713 · 5524403 · 9530429 · 15608017 · 18453683327)</td><td></td><td></td><td></td></tr><tr><td>77, 139 + (194dd)</td><td>29819053207472296379159807374330473560010495877 (47dd)</td><td>Z/6Z</td><td>220121</td></tr><tr><td>(2) × (2 · 3 · 19 · 101 · 103 · 1787 · 1590643 · 13909117 · 29656051 · 39410677 · 272067307)</td><td></td><td></td><td></td></tr><tr><td>92, 137 + (231dd)</td><td>6199291385330615425337006319139287283518435008717 (49dd)</td><td>Z/6Z</td><td>220122</td></tr><tr><td>(2<sup>2</sup> · 3<sup>5</sup> · 5<sup>2</sup> · 103 · 1889 · 4073 · 203617 · 293507 · 531667 · 19257083 · 526125653951)</td><td></td><td></td><td></td></tr><tr><td>70, 149 - (223dd)</td><td>24721694635072630263761573923973931287719915271 (47dd)</td><td>Z/6Z</td><td>220123</td></tr><tr><td>(2) × (2<sup>2</sup> · 3<sup>6</sup> · 11 · 89 · 227 · 443 · 33749 · 219353 · 8522959 · 35764961 · 19080765961)</td><td></td><td></td><td></td></tr><tr><td>82, 134 + (234dd)</td><td>2308832612697750858892228636505628255991181169289 (49dd)</td><td>Z/6Z</td><td>220123</td></tr><tr><td>(2<sup>4</sup> · 3<sup>2</sup> · 29 · 16001 · 48187 · 107699 · 7847317 · 8143147 · 15948781 · 6532844293)</td><td></td><td></td><td></td></tr><tr><td>71, 139 + (248dd)</td><td>33836890734036629261924516665840404273381301 (44dd)</td><td>Z/6Z</td><td>220124</td></tr><tr><td>(2<sup>2</sup> · 3) × (2<sup>4</sup> · 3 · 5 · 619 · 761 · 39719 · 639959 · 1003411 · 12898657 · 75813901489)</td><td></td><td></td><td></td></tr></tbody></table>
<table><thead><tr><th>N factor of (dd)</th><th>p | N(dd)</th><th>family</th><th>when</th></tr></thead><tbody><tr><td>69, 139 + (166dd)</td><td>7932688902674767877319906826600442482951 (41dd)<br/>&lt;2&sup3; &times; 2&sup3; &middot; 3 &middot; 14699 &middot; 37591 &middot; 1196869 &middot; 2181937 &middot; 8831281 &middot; 129686565067&gt;</td><td>S<sub>11</sub></td><td>220114</td></tr><tr><td>52, 149 + (168dd)</td><td>69893670127458726482916735173225009349257 (41dd)<br/>&lt;2&sup2; &times; 2&sup3; &middot; 59 &middot; 103 &middot; 293 &middot; 1699 &middot; 1747 &middot; 4933 &middot; 909031 &middot; 1620469 &middot; 56874054581&gt;</td><td>S<sub>11</sub></td><td>220114</td></tr><tr><td>79, 139 + (171dd)</td><td>596931162601751923425190492553980981393 (39dd)<br/>&lt;2&sup3; &times; 2&sup3; &middot; 7&sup2; &middot; 269&sup2; &middot; 1721 &middot; 19543 &middot; 246683 &middot; 440239 &middot; 720184550543&gt;</td><td>S<sub>11</sub></td><td>220114</td></tr><tr><td>96, 142 + (192dd)</td><td>2191432596178567828861398613285221 (35dd)<br/>&lt;2&sup2; &middot; 3 &middot; 7 &middot; 47 &middot; 229 &middot; 12343 &middot; 8053337 &middot; 15452729 &middot; 15780243509&gt;</td><td>S<sub>11</sub></td><td>220114</td></tr><tr><td>72, 149 - (187dd)</td><td>315326499129576793697964957827057 (33dd)<br/>&lt;2&sup2; &middot; 3&sup3; &middot; 11 &middot; 31 &middot; 307 &middot; 1117 &middot; 9733 &middot; 85829 &middot; 980957 &middot; 30469129&gt;</td><td>S<sub>11</sub></td><td>220114</td></tr><tr><td>72, 149 - (155dd)</td><td>357142978908396162737234145844921 (33dd)<br/>&lt;2&sup2; &times; 2 &middot; 31 &middot; 7867 &middot; 136777 &middot; 2048327 &middot; 7010291 &middot; 186407281&gt;</td><td>S<sub>11</sub></td><td>220114</td></tr><tr><td>93, 137 - (197dd)</td><td>223183212056174382934792667 (27dd)<br/>&lt;2&sup5; &middot; 13 &middot; 1621 &middot; 49033 &middot; 1982447 &middot; 3404827603&gt;</td><td>S<sub>11</sub></td><td>220114</td></tr><tr><td>86, 134 + (186dd)</td><td>49658691102011202484249214533 (29dd)<br/>&lt;2&sup2; &times; 2&sup4; &middot; 3 &middot; 5 &middot; 7 &middot; 23 &middot; 79 &middot; 151 &middot; 587 &middot; 2591 &middot; 39509 &middot; 896442761&gt;</td><td>S<sub>11</sub></td><td>220114</td></tr><tr><td>86, 145 + (164dd)</td><td>191954653450671186092815000481 (30dd)<br/>&lt;2&sup3; &middot; 3&sup2; &middot; 179 &middot; 5573 &middot; 389461 &middot; 530177 &middot; 12943127117&gt;</td><td>S<sub>11</sub></td><td>220114</td></tr><tr><td>82, 146 + (198dd)</td><td>120132845434010252906182780753 (30dd)<br/>&lt;2&sup2; &middot; 887 &middot; 10067 &middot; 26813 &middot; 1767407 &middot; 70973480791&gt;</td><td>S<sub>11</sub></td><td>220114</td></tr><tr><td>89, 131 + (207dd)</td><td>35636694942516749949880764536158589 (35dd)<br/>&lt;2&sup2; &times; 2&sup3; &middot; 11 &middot; 17 &middot; 61 &middot; 89 &middot; 17029 &middot; 629467 &middot; 1321079 &middot; 77463138941&gt;</td><td>S<sub>11</sub></td><td>220115</td></tr><tr><td>74, 139 + (205dd)</td><td>59091747801762858535122297401611962258962823803 (47dd)<br/>&lt;2&sup3; &middot; 3 &middot; 23 &middot; 141991 &middot; 2297863 &middot; 4360033 &middot; 12672377 &middot; 25491761 &middot; 232945920323&gt;</td><td>S<sub>11</sub></td><td>220115</td></tr><tr><td>89, 131 + (173dd)</td><td>15673997760117375613229343840498013333349 (41dd)<br/>&lt;2&sup2; &times; 2&sup3; &middot; 61 &middot; 739 &middot; 52609 &middot; 1395077 &middot; 1542589 &middot; 12281407 &middot; 7814459579&gt;</td><td>S<sub>11</sub></td><td>220115</td></tr><tr><td>84, 134 + (224dd)</td><td>42401931560000268689457878037331393 (35dd)<br/>&lt;2&sup4; &middot; 3 &middot; 19 &middot; 521 &middot; 1367 &middot; 108139 &middot; 230833 &middot; 5931799 &middot; 440877457&gt;</td><td>S<sub>11</sub></td><td>220116</td></tr><tr><td>69, 139 - (191dd)</td><td>17987049449879167615904106068499621810958643 (44dd)<br/>&lt;2&sup2; &times; 2&sup3; &middot; 11&sup2; &middot; 199 &middot; 2287 &middot; 36433 &middot; 596081 &middot; 7710853 &middot; 21111049 &middot; 5774613001&gt;</td><td>S<sub>11</sub></td><td>220116</td></tr><tr><td>85, 134 + (213dd)</td><td>64480050612882732136749348406628183869 (38dd)<br/>&lt;2&middot;3 &times; 2&middot;3&sup2; &middot; 251 &middot; 21269 &middot; 3369409 &middot; 11717963 &middot; 13937953 &middot; 203224457&gt;</td><td>S<sub>11</sub></td><td>220116</td></tr><tr><td>58, 149 + (222dd)</td><td>414001338870756676319995466729941 (33dd)<br/>&lt;2&sup2; &times; 2&sup3; &middot; 3 &middot; 7&sup2; &middot; 12527 &middot; 3846841 &middot; 20845273 &middot; 175228816369&gt;</td><td>S<sub>11</sub></td><td>220116</td></tr><tr><td>68, 149 + (212dd)</td><td>86480478473510065337924664955072005883 (38dd)<br/>&lt;2&sup2; &times; 2&sup5; &middot; 3 &middot; 13859 &middot; 3597569 &middot; 5447051 &middot; 24276271 &middot; 68317569857&gt;</td><td>S<sub>11</sub></td><td>220116</td></tr><tr><td>74, 149 + (217dd)</td><td>4113057328315019941574284479 (29dd)<br/>&lt;2&sup2; &middot; 3 &middot; 71 &middot; 45061 &middot; 271753 &middot; 2102207 &middot; 1875316253&gt;</td><td>S<sub>11</sub></td><td>220116</td></tr><tr><td>82, 149 - (218dd)</td><td>1847780467621495734449284245985360331 (37dd)<br/>&lt;2&sup7; &middot; 3 &middot; 2539 &middot; 25349 &middot; 93889 &middot; 673349 &middot; 12534209 &middot; 94350401&gt;</td><td>S<sub>11</sub></td><td>220116</td></tr><tr><td>82, 139 - (230dd)</td><td>6983643688295927277236975661223389491 (37dd)<br/>&lt;2&sup2; &times; 2&sup5; &middot; 3&sup2; &middot; 1723 &middot; 3703109 &middot; 6993839 &middot; 23121011 &middot; 11751285181&gt;</td><td>S<sub>11</sub></td><td>220116</td></tr><tr><td>75, 142 + (216dd)</td><td>6589446638092599241138023917 (28dd)<br/>&lt;2&sup6; &middot; 29 &middot; 9377 &middot; 34513 &middot; 198479 &middot; 55272585847&gt;</td><td>S<sub>11</sub></td><td>220116</td></tr><tr><td>69, 139 - (224dd)</td><td>313626806309044386261774730369151 (33dd)<br/>&lt;2&sup4; &middot; 3 &middot; 5 &middot; 79 &middot; 311 &middot; 2129 &middot; 11369 &middot; 548441 &middot; 757481 &middot; 5289503&gt;</td><td>S<sub>11</sub></td><td>220116</td></tr><tr><td>97, 149 + (215dd)</td><td>33182756619242790311549689795255613 (35dd)<br/>&lt;2&sup2; &middot; 3 &middot; 5 &middot; 11&sup2; &middot; 139 &middot; 181 &middot; 197 &middot; 311 &middot; 20593 &middot; 1063441 &middot; 135401299063&gt;</td><td>S<sub>11</sub></td><td>220116</td></tr><tr><td>74, 137 + (227dd)</td><td>2048174726214184971516279798069913 (34dd)<br/>&lt;2&sup2; &times; 2&sup5; &middot; 3 &middot; 941 &middot; 43961 &middot; 557759 &middot; 8667961 &middot; 53339022479&gt;</td><td>S<sub>11</sub></td><td>220116</td></tr><tr><td>99, 146 + (231dd)</td><td>109656245683327340782944722989 (30dd)<br/>&lt;2&sup2; &middot; 5 &middot; 4019 &middot; 44623 &middot; 113489 &middot; 224359 &middot; 1200685709&gt;</td><td>S<sub>11</sub></td><td>220116</td></tr><tr><td>84, 139 - (227dd)</td><td>124162480994347566248707849347947447 (35dd)<br/>&lt;2&sup4; &middot; 7 &middot; 13 &middot; 29 &middot; 41 &middot; 48259 &middot; 678421 &middot; 1344509 &middot; 162931793303&gt;</td><td>S<sub>11</sub></td><td>220116</td></tr><tr><td>84, 139 - (193dd)</td><td>127557569748786523440396285491916792547 (39dd)<br/>&lt;2&sup3; &middot; 3&sup3; &middot; 73 &middot; 197 &middot; 283 &middot; 10301 &middot; 21613 &middot; 6510073 &middot; 7646179 &middot; 13093393&gt;</td><td>S<sub>11</sub></td><td>220116</td></tr><tr><td>88, 137 + (231dd)</td><td>17986962943353259149414934888573541222161 (41dd)<br/>&lt;2&middot;3 &times; 2&sup3; &middot; 3 &middot; 19 &middot; 23&sup2; &middot; 37 &middot; 389 &middot; 401 &middot; 57223 &middot; 3302939 &middot; 8434963 &middot; 1350628841&gt;</td><td>S<sub>11</sub></td><td>220116</td></tr><tr><td>53, 149 + (216dd)</td><td>49824306994528881877410440109535859298766931 (44dd)<br/>&lt;2&sup3; &middot; 3&sup2; &middot; 2671 &middot; 18869 &middot; 40499 &middot; 4033853 &middot; 5213777 &middot; 11264959 &middot; 1430999767&gt;</td><td>S<sub>11</sub></td><td>220116</td></tr><tr><td>77, 139 - (217dd)</td><td>2913947035781063821596011687931095464368209 (43dd)<br/>&lt;2&sup4; &middot; 5 &middot; 7 &middot; 1823 &middot; 4231 &middot; 24359 &middot; 159431 &middot; 2112191 &middot; 44658907 &middot; 1841579321&gt;</td><td>S<sub>11</sub></td><td>220116</td></tr><tr><td>84, 142 + (219dd)</td><td>271893653179988144731484255881181816283712096681 (48dd)<br/>&lt;2&sup2; &times; 2 &middot; 7 &middot; 137 &middot; 607 &middot; 3853 &middot; 1069273 &middot; 2749679 &middot; 7486327 &middot; 8924701 &middot; 154276474453&gt;</td><td>S<sub>11</sub></td><td>220116</td></tr><tr><td>87, 139 + (225dd)</td><td>205388471386709230099710142427317022684953 (42dd)<br/>&lt;2&sup2; &middot; 17 &middot; 23 &middot; 83 &middot; 887 &middot; 1487 &middot; 10529 &middot; 1328477 &middot; 1854709 &middot; 3696557 &middot; 12508703&gt;</td><td>S<sub>11</sub></td><td>220116</td></tr><tr><td>83, 149 + (214dd)</td><td>723577182616554422038709801648887 (33dd)<br/>&lt;2&sup2; &times; 2 &middot; 31 &middot; 4759 &middot; 340931 &middot; 464747 &middot; 16010627 &middot; 241671841&gt;</td><td>S<sub>11</sub></td><td>220116</td></tr><tr><td>71, 149 + (231dd)</td><td>1327578397518109970479951051 (28dd)<br/>&lt;2&sup2; &times; 2&sup2; &middot; 5&sup3; &middot; 7&sup2; &middot; 43 &middot; 223 &middot; 12931007 &middot; 218503516873&gt;</td><td>S<sub>11</sub></td><td>220116</td></tr></tbody></table>
<table><thead><tr><th>N factor of (dd)</th><th>p | N(dd)</th><th>family</th><th>when</th></tr></thead><tbody><tr><td>54, 149 + (228dd)</td><td>1525947144376237945730724897659097491 (37dd)<br/>&lt;2<sup>6</sup> . 3 . 71 . 33797 . 74383 . 1415221 . 39335017 . 799879313&gt;</td><td>S<sub>11</sub></td><td>220116</td></tr><tr><td>54, 149 + (192dd)</td><td>24982548838086157051576308905145911958421778531 (47dd)<br/>&lt;2<sup>3</sup> . 3<sup>3</sup> . 5 . 277 . 60013 . 179327 . 943357 . 4542001 . 13200221 . 137194814203&gt;</td><td>S<sub>11</sub></td><td>220116</td></tr><tr><td>92, 131 - (231dd)</td><td>5294940733976016020635573563645720161 (37dd)<br/>&lt;2<sup>2</sup> . 5<sup>2</sup> . 13<sup>2</sup> . 37 . 43 . 137 . 919 . 12281 . 21491 . 77369 . 76596830881&gt;</td><td>S<sub>11</sub></td><td>220116</td></tr><tr><td>67, 149 - (230dd)</td><td>913701696563975714033091493225254422205593287 (45dd)<br/>&lt;2&gt; &times; &lt;2<sup>6</sup> . 5<sup>3</sup> . 7 . 13 . 67 . 521 . 887 . 38977 . 79393 . 79399 . 5144093 . 16035866953&gt;</td><td>S<sub>11</sub></td><td>220116</td></tr><tr><td>79, 137 + (228dd)</td><td>1550858839575481986003226594036035233975681 (44dd)<br/>&lt;2&gt; &times; &lt;2<sup>4</sup> . 3 . 19 . 3307 . 22817 . 24329 . 37097 . 3687401 . 20724251 . 1633775719&gt;</td><td>S<sub>11</sub></td><td>220116</td></tr><tr><td>87, 139 + (184dd)</td><td>3746270639283330432738264682702011658979991767 (46dd)<br/>&lt;2<sup>5</sup> . 3<sup>2</sup> . 79 . 271 . 25541 . 320317 . 414101 . 873343 . 2684753 . 76488639113&gt;</td><td>S<sub>11</sub></td><td>220116</td></tr><tr><td>92, 137 - (229dd)</td><td>68741933196114943137300875603 (29dd)<br/>&lt;2<sup>4</sup> . 3 . 5<sup>2</sup> . 37 . 157 . 677 . 23677 . 131321 . 4684784249&gt;</td><td>S<sub>11</sub></td><td>220116</td></tr><tr><td>80, 137 - (239dd)</td><td>764876847089550595623709963544321 (33dd)<br/>&lt;2&gt; &times; &lt;2<sup>12</sup> . 11 . 13 . 41 . 109 . 251 . 277 . 457 . 173023 . 26575512637&gt;</td><td>S<sub>11</sub></td><td>220117</td></tr><tr><td>76, 139 - (239dd)</td><td>2509783409036652433415112517561 (31dd)<br/>&lt;2<sup>3</sup> . 23 . 353653 . 595207 . 925051 . 70049875451&gt;</td><td>S<sub>11</sub></td><td>220117</td></tr><tr><td>68, 142 + (197dd)</td><td>31652779243310772656750398477 (29dd)<br/>&lt;2&gt; &times; &lt;2<sup>5</sup> . 5 . 37 . 151 . 1215463 . 4887709 . 2980136213&gt;</td><td>S<sub>11</sub></td><td>220117</td></tr><tr><td>80, 139 + (213dd)</td><td>628775525747747428878554121326080661368988704367 (48dd)<br/>&lt;2<sup>3</sup> . 7<sup>2</sup> . 11 . 41 . 347 . 839 . 8161 . 113623 . 1912553 . 4209991 . 16022309 . 102120329&gt;</td><td>S<sub>11</sub></td><td>220117</td></tr><tr><td>87, 149 + (238dd)</td><td>334811947595854120996770967 (27dd)<br/>&lt;2<sup>8</sup> . 127 . 163 . 653 . 14540863 . 6653747141&gt;</td><td>S<sub>11</sub></td><td>220117</td></tr><tr><td>68, 142 + (231dd)</td><td>13823119660712662008022831172622217 (35dd)<br/>&lt;2<sup>5</sup> . 5 . 727 . 26141 . 42841 . 846067 . 2793179 . 44902069&gt;</td><td>S<sub>11</sub></td><td>220117</td></tr><tr><td>73, 142 + (239dd)</td><td>10098261615225106587019034717 (29dd)<br/>&lt;2<sup>4</sup> . 89 . 263 . 383 . 125959 . 5080237 . 110019313&gt;</td><td>S<sub>11</sub></td><td>220117</td></tr><tr><td>73, 142 + (211dd)</td><td>4654023020427626964071882457961055105609 (40dd)<br/>&lt;2<sup>2</sup>&gt; &times; &lt;2<sup>2</sup> . 3 . 109 . 1279 . 8971 . 78179 . 14512357 . 29777023 . 2294774683&gt;</td><td>S<sub>11</sub></td><td>220117</td></tr><tr><td>87, 134 + (242dd)</td><td>98174636130339458381137281532902315113 (38dd)<br/>&lt;2<sup>3</sup> . 3<sup>2</sup> . 11 . 31 . 241 . 1171 . 10477 . 288293 . 15861457 . 295749737377&gt;</td><td>S<sub>11</sub></td><td>220117</td></tr><tr><td>52, 149 - (238dd)</td><td>9504145515920220202657769053720536529 (37dd)<br/>&lt;2<sup>2</sup> . 19 . 23 . 107 . 166843 . 647719 . 3440077 . 4746431 . 28797707&gt;</td><td>S<sub>11</sub></td><td>220117</td></tr><tr><td>63, 142 + (243dd)</td><td>499347379550181541744752317269 (30dd)<br/>&lt;2&gt; &times; &lt;2 . 3 . 197 . 277 . 25127 . 71941 . 7945849 . 53090659&gt;</td><td>S<sub>11</sub></td><td>220117</td></tr><tr><td>91, 139 + (244dd)</td><td>11491948197434214683927366021139960467983744846534603 (54dd)<br/>&lt;2&gt; &times; &lt;2<sup>4</sup> . 31 . 79 . 173 . 919 . 73369 . 94781 . 630919 . 1291117 . 1337363 . 7448509 . 163456253&gt;</td><td>S<sub>11</sub></td><td>220117</td></tr><tr><td>65, 142 + (232dd)</td><td>16543228921610947547423319808009757592543469 (44dd)<br/>&lt;3&gt; &times; &lt;2<sup>2</sup> . 3<sup>2</sup> . 7 . 23 . 23357 . 83689 . 205097 . 254773 . 35704729 . 260883587927&gt;</td><td>S<sub>11</sub></td><td>220117</td></tr><tr><td>76, 139 + (244dd)</td><td>516149857905663948423972078277609881449 (39dd)<br/>&lt;2<sup>2</sup> . 3<sup>2</sup> . 7 . 457 . 701 . 1109 . 1609 . 77269 . 2087909 . 3787097 . 5864489&gt;</td><td>S<sub>11</sub></td><td>220117</td></tr><tr><td>83, 134 + (136dd)</td><td>5353720031116679424387066050214337 (34dd)<br/>&lt;2<sup>2</sup> . 11 . 23 . 251 . 337 . 79397 . 1012507 . 3820319 . 203643131&gt;</td><td>S<sub>11</sub></td><td>220117</td></tr><tr><td>83, 134 + (239dd)</td><td>91002962967364396509632176333 (29dd)<br/>&lt;2<sup>2</sup> . 3 . 7 . 13 . 75353 . 84533 . 1707707 . 7661127787&gt;</td><td>S<sub>11</sub></td><td>220117</td></tr><tr><td>68, 142 + (169dd)</td><td>19468423591945940785265108842305714323526289 (44dd)<br/>&lt;2&gt; &times; &lt;2 . 3 . 7 . 13 . 449 . 21383 . 78839 . 359027 . 3411649 . 17114429 . 1123564987&gt;</td><td>S<sub>11</sub></td><td>220117</td></tr><tr><td>90, 139 + (241dd)</td><td>2150378303491644833157192790279927 (34dd)<br/>&lt;2<sup>3</sup> . 199 . 1269013 . 9441227 . 9795209 . 11509688987&gt;</td><td>S<sub>11</sub></td><td>220117</td></tr><tr><td>83, 134 + (170dd)</td><td>2324888459241523990334435780743469 (34dd)<br/>&lt;2&gt; &times; &lt;2<sup>3</sup> . 5 . 1709 . 7933 . 1504991 . 8027849 . 177418751927&gt;</td><td>S<sub>11</sub></td><td>220117</td></tr><tr><td>83, 139 + (233dd)</td><td>36625164225110586401367754245157 (32dd)<br/>&lt;2<sup>2</sup> . 5 . 37 . 1231 . 3041 . 146749 . 20461193 . 4403188577&gt;</td><td>S<sub>11</sub></td><td>220117</td></tr><tr><td>86, 139 - (244dd)</td><td>5573224213511683912426525754466119318810153 (43dd)<br/>&lt;2<sup>5</sup> . 571 . 5443 . 29429 . 864307 . 10488781 . 13454213 . 15611881367&gt;</td><td>S<sub>11</sub></td><td>220117</td></tr><tr><td>95, 142 + (243dd)</td><td>10085922450540433565244186723577 (32dd)<br/>&lt;2&gt; &times; &lt;2<sup>7</sup> . 131 . 137 . 271 . 32693 . 488321 . 507404517889&gt;</td><td>S<sub>11</sub></td><td>220117</td></tr><tr><td>62, 146 + (236dd)</td><td>71683364969105361862733339919966149 (35dd)<br/>&lt;2&gt; &times; &lt;2<sup>5</sup> . 2683 . 15493 . 80537 . 320317 . 18080459 . 36972371&gt;</td><td>S<sub>11</sub></td><td>220117</td></tr><tr><td>80, 139 + (241dd)</td><td>8183498546729728789676899241 (28dd)<br/>&lt;2<sup>3</sup> . 103 . 337 . 3191 . 14629 . 41117 . 60091 . 255511&gt;</td><td>S<sub>11</sub></td><td>220117</td></tr><tr><td>68, 142 + (126dd)</td><td>3011632726083334175010295234518818178029 (40dd)<br/>&lt;2&gt; &times; &lt;2 . 11 . 59 . 73 . 107 . 971 . 383869 . 1384741 . 21810781 . 13193154319&gt;</td><td>S<sub>11</sub></td><td>220117</td></tr><tr><td>83, 134 + (210dd)</td><td>7937310628273890338230982131430409871673 (40dd)<br/>&lt;2<sup>4</sup> . 13 . 97 . 647 . 40471 . 224897 . 8848453 . 18041563 . 418470571&gt;</td><td>S<sub>11</sub></td><td>220117</td></tr><tr><td>91, 131 + (248dd)</td><td>2329966057028384009201034265975365952286201 (43dd)<br/>&lt;2<sup>3</sup>&gt; &times; &lt;2<sup>6</sup> . 821 . 1499 . 3581 . 92671 . 649511 . 41957147 . 408878441767&gt;</td><td>S<sub>11</sub></td><td>220118</td></tr><tr><td>63, 142 + (214dd)</td><td>15949743522706797612257367166134770473 (38dd)<br/>&lt;2&gt; &times; &lt;2 . 102071 . 619741 . 7887511 . 36246611 . 220482299371&gt;</td><td>S<sub>11</sub></td><td>220118</td></tr></tbody></table>
<table><thead><tr><th>N factor of (dd)</th><th>p | N(dd)</th><th>family</th><th>when</th></tr></thead><tbody><tr><td>87, 134 + (204dd)</td><td>34202454020527149377831516484061980645737 (41dd)</td><td>S<sub>11</sub></td><td>220118</td></tr><tr><td>&lt;2<sup>2</sup>&gt; &times; &lt;2<sup>4</sup>&middot;3&middot;5<sup>3</sup>&middot;41&middot;73&middot;79&middot;191&middot;631&middot;10331&middot;39229&middot;43445371&middot;2840248781&gt;</td><td></td><td></td><td></td></tr><tr><td>91, 139 - (250dd)</td><td>4519483920649446786007684574183 (31dd)</td><td>S<sub>11</sub></td><td>220118</td></tr><tr><td>&lt;2&gt; &times; &lt;2<sup>2</sup>&middot;3&middot;13&middot;787&middot;4441&middot;296503&middot;19074481&middot;732819343&gt;</td><td></td><td></td><td></td></tr><tr><td>96, 134 + (247dd)</td><td>1624606283502856081380875678929 (31dd)</td><td>S<sub>11</sub></td><td>220118</td></tr><tr><td>&lt;2<sup>8</sup>&middot;11&middot;1103&middot;115783&middot;15469073&middot;292032293071&gt;</td><td></td><td></td><td></td></tr><tr><td>94, 134 + (248dd)</td><td>1974464276958891396660765763865448049 (37dd)</td><td>S<sub>11</sub></td><td>220118</td></tr><tr><td>&lt;2&gt; &times; &lt;2&middot;3&middot;7&middot;3041&middot;1024159&middot;2604559&middot;27283591&middot;106206343807&gt;</td><td></td><td></td><td></td></tr><tr><td>94, 137 - (247dd)</td><td>9870926313672469907167292077290083550749 (40dd)</td><td>S<sub>11</sub></td><td>220118</td></tr><tr><td>&lt;2<sup>2</sup>&middot;13&middot;29&middot;31&middot;7069&middot;3328993&middot;14536441&middot;45097501&middot;13687163153&gt;</td><td></td><td></td><td></td></tr><tr><td>55, 149 + (250dd)</td><td>8974951723535129435403638112332221 (34dd)</td><td>S<sub>11</sub></td><td>220118</td></tr><tr><td>&lt;2<sup>4</sup>&middot;11&middot;3511&middot;7687&middot;1961929&middot;8642863&middot;111427099859&gt;</td><td></td><td></td><td></td></tr><tr><td>94, 131 + (250dd)</td><td>1237726513000116999529829161 (28dd)</td><td>S<sub>11</sub></td><td>220118</td></tr><tr><td>&lt;2&middot;3&gt; &times; &lt;2&middot;3&middot;2591&middot;760231&middot;5339311&middot;3269068247&gt;</td><td></td><td></td><td></td></tr><tr><td>88, 134 + (245dd)</td><td>1685406455599302200184201109 (28dd)</td><td>S<sub>11</sub></td><td>220118</td></tr><tr><td>&lt;2<sup>2</sup>&gt; &times; &lt;2<sup>3</sup>&middot;4049&middot;77267&middot;6534467&middot;25763370461&gt;</td><td></td><td></td><td></td></tr></tbody></table>