nyuuzyou/wikis huggingface dataset:
383

== Number ==
$383$ (three hundred and eighty-three) is:
:The $76$th prime number
:The $6$th Woodall number after $1$, $7$, $23$, $63$, $159$, and $3$rd Woodall prime after $7$, $23$:
::$383 = 6 \times 2^6 - 1$
:The $7$th Thabit number after $(2)$, $5$, $11$, $23$, $47$, $95$, $191$:
::$383 = 3 \times 2^7 - 1$ 
:The $7$th Thabit prime after $2$, $5$, $11$, $23$, $47$, $191$:
::$383 = 3 \times 2^7 - 1$ 
:The $14$th palindromic prime:
::$2$, $3$, $5$, $7$, $11$, $101$, $131$, $151$, $181$, $191$, $313$, $353$, $373$, $383$, $\ldots$
:The $15$th safe prime after $5$, $7$, $11$, $23$, $47$, $59$, $83$, $107$, $167$, $179$, $227$, $263$, $347$, $359$:
::$383 = 2 \times 191 + 1$
:The $20$th near-repdigit prime after $101$, $113$, $131$, $151$, $181$, $191$, $199$, $211$, $223$, $227$, $229$, $233$, $277$, $311$, $313$, $331$, $337$, $353$, $373$
:The $20$th inconsummate number after $62$, $63$, $65$, $75$, $84$, $95$, $161$, $173$, $195$, $216$, $261$, $266$, $272$, $276$, $326$, $371$, $372$, $377$, $381$:
::$\nexists n \in \Z_{>0}: n = 383 \times \map {s_{10} } n$
:The $28$th long period prime after $7$, $17$, $19$, $23$, $29$, $\ldots$, $181$, $193$, $223$, $229$, $233$, $257$, $263$, $269$, $313$, $337$, $367$, $379$
:The $30$th left-truncatable prime after $2$, $3$, $5$, $7$, $\ldots$, $197$, $223$, $283$, $313$, $317$, $337$, $347$, $353$, $367$, $373$
== Also see ==
* 
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Category:Specific Numbers
Category:383
384

== Number ==
$384$ (three hundred and eighty-four) is:
:$2^7 \times 3$
:The $6$th positive integer after $128$, $192$, $256$, $288$, $320$ with $7$ or more prime factors:
::$384 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \, \paren {\times \, 3}$
:The index (after $2$, $3$, $6$, $30$, $75$, $81$, $115$, $123$, $249$, $362$) of the $11$th Woodall prime:
::$384 \times 2^{384} - 1$
:The $17$th positive integer after $64$, $96$, $128$, $144$, $160$, $192$, $216$, $224$, $240$, $256$, $288$, $320$, $324$, $336$, $352$, $360$ with $6$ or more prime factors:
::$384 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \, \paren {\times \, 2 \times 3}$
:The $28$th Zuckerman number after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $11$, $12$, $\ldots$, $135$, $144$, $175$, $212$, $216$, $224$, $312$, $315$:
::$384 = 4 \times 96 = 4 \times \paren {3 \times 8 \times 4}$
== Also see ==
* 
* 
* 
* 
Category:Specific Numbers
Category:384
3844

== Number ==
$3844$ (three thousand, eight hundred and forty-four) is:
:$2^2 \times 31^2$
:The $23$rd square number after $1$, $4$, $36$, $121$, $144$, $256$, $\ldots$, $1936$, $2304$, $2704$, $2916$, $3136$, $3600$ to be the divisor sum value of some (strictly) positive integer:
::$3844 = \map {\sigma_1} {1200}$
:The $62$nd square number after $1$, $4$, $9$, $16$, $25$, $36$, $\ldots$, $3249$, $3364$, $3481$, $3600$, $3721$:
::$3844 = 62 \times 62$
== Also see ==
* 
* 
Category:Specific Numbers
Category:3844
385

== Number ==
$385$ (three hundred and eighty-five) is:
:$5 \times 7 \times 11$
:The $3$rd of the $3$-digit integers $m$ which need the largest number of reverse-and-add process iterations ($23$) before reaching a palindromic number:
::$385$, $968$, $1837$, $\ldots$, $8713200023178$
:The $7$th Cullen number after $1$, $3$, $9$, $25$, $65$, $161$:
::$385 = 6 \times 2^6 + 1$
:The $10$th square pyramidal number after $1$, $5$, $14$, $30$, $55$, $91$, $140$, $204$, $285$:
::$385 = \ds \sum_{k \mathop = 1}^{10} k^2 = \dfrac {10 \paren {10 + 1} \paren {2 \times 10 + 1} } 6$
:The number of integer partitions for $18$:
::$\map p {18} = 385$
:The $19$th positive integer $n$ after $0$, $1$, $5$, $25$, $29$, $41$, $49$, $61$, $65$, $85$, $89$, $101$, $125$, $145$, $149$, $245$, $265$, $365$ such that the Fibonacci number $F_n$ ends in $n$
:The $23$rd number whose divisor sum is square:
:
:The $42$nd sphenic number after $30$, $42$, $66$, $70$, $\ldots$, $290$, $310$, $318$, $322$, $345$, $354$, $357$, $366$, $370$, $374$:
::$385 = 5 \times 7 \times 11$
== Also see ==
* 
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* 
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* 
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* 
Category:Specific Numbers
Category:385
386

== Number ==
$386$ (three hundred and eighty-six) is:
:$2 \times 193$
:The $39$th noncototient after $10$, $26$, $34$, $50$, $\ldots$, $290$, $292$, $298$, $310$, $326$, $340$, $344$, $346$, $362$, $366$, $372$:
::$\nexists m \in \Z_{>0}: m - \phi \left({m}\right) = 386$
:where $\phi \left({m}\right)$ denotes the Euler $\phi$ function
:The $21$st inconsummate number after $62$, $63$, $65$, $75$, $84$, $95$, $161$, $173$, $195$, $216$, $261$, $266$, $272$, $276$, $326$, $371$, $372$, $377$, $381$, $383$:
::$\nexists n \in \Z_{>0}: n = 386 \times s_{10} \left({n}\right)$
== Also see ==
* 
* 
Category:Specific Numbers
Category:386
387

== Number ==
$387$ (three hundred and eighty-seven) is:
:$3^2 \times 43$
:The $22$nd inconsummate number after $62$, $63$, $65$, $75$, $\ldots$, $371$, $372$, $377$, $381$, $383$, $386$:
::$\nexists n \in \Z_{>0}: n = 387 \map {\times s_{10} } n$
:The $1$st of the $11$th pair of consecutive integers which both have $6$ divisors:
::$\map {\sigma_0} {387} = \map {\sigma_0} {388} = 6$
== Also see ==
* 
* 
Category:Specific Numbers
Category:387
3870

== Number ==
$3870$ (three thousand, eight hundred and seventy) is:
:$2 \times 3^2 \times 5 \times 43$
: The $4$th integer after $954$, $1980$, $2961$ whose digits are unchanged when subtracting its reversal:
::$3870 - 0783 = 3087$
== Also see ==
* 
Category:Specific Numbers
Category:3870
388

== Number ==
$388$ (three hundred and eighty-eight) is:
:$2^2 \times 97$
:The $6$th positive integer after $1$, $24$, $26$, $87$, $168$ whose Euler $\phi$ value is equal to the product of its digits: 
::$\map \phi {388} = 192 = 3 \times 8 \times 8$
:The $2$nd of the $11$th pair of consecutive integers which both have $6$ divisors:
::$\map {\sigma_0} {387} = \map {\sigma_0} {388} = 6$
== Also see ==
* 
* 
Category:Specific Numbers
Category:388
389

== Number ==
$389$ (three hundred and eighty-nine) is:
:The $77$th prime number
:The smaller of the $3$rd pair of primes whose prime gap is $8$:
::$397 - 389 = 8$
:The $20$th emirp after $13$, $17$, $31$, $37$, $71$, $73$, $79$, $97$, $107$, $113$, $149$, $157$, $167$, $179$, $199$, $311$, $337$, $347$, $359$
:The $29$th long period prime after $7$, $17$, $19$, $23$, $29$, $\ldots$, $269$, $313$, $337$, $367$, $379$, $383$
== Also see ==
* 
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* 
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Category:Specific Numbers
Category:389
39

== Number ==
$39$ (thirty-nine) is:
:$3 \times 13$
:The smallest positive integer having a multiplicative persistence of $3$.
:The $15$th semiprime after $4$, $6$, $9$, $10$, $14$, $15$, $21$, $22$, $25$, $26$, $33$, $34$, $35$, $38$:
::$39 = 3 \times 13$
:The $21$st after $1$, $2$, $4$, $5$, $6$, $8$, $9$, $12$, $13$, $15$, $16$, $17$, $20$, $24$, $25$, $27$, $28$, $32$, $35$, $26$ of the $24$ positive integers which cannot be expressed as the sum of distinct non-pythagorean primes.
:The $28$th integer $n$ such that $2^n$ contains no zero in its decimal representation:
::$2^{39} = 549 \, 755 \, 813 \, 888$
:The number of convex polygons that can be assembled from the complete set of $12$ hexiamonds.
:Suggested by in his $1986$ book as being the smallest uninteresting number, which fact makes it intrinsically interesting.
== Also see ==
* Interesting Number Paradox
* 
* 
* 
* 
== Sources ==
* : $39$
Category:Specific Numbers
Category:39
39,993

== Number ==
$39 \, 993$ (thirty-nine thousand, nine hundred and ninety-three) is:
:$3 \times 13 \, 331$
:The $15$th integer after $0$, $1$, $3$, $5$, $7$, $9$, $33$, $99$, $313$, $585$, $717$, $7447$, $15 \, 351$, $32 \, 223$ which is palindromic in both decimal and binary:
::$39 \, 993_{10} = 1 \, 001 \, 110 \, 000 \, 111 \, 001_2$
== Also see ==
* 
Category:Specific Numbers
Category:39,993
390,625

== Number ==
$390 \, 625$ (three hundred and ninety thousand, six hundred and twenty-five) is:
:$5^8$
:The $5$th eighth power after $1$, $256$, $6561$, $65 \, 536$:
::$390 \, 625 = 5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5$
:The $8$th power of $5$ after $(1)$, $5$, $25$, $125$, $625$, $3125$, $15 \, 625$, $78 \, 125$:
::$390 \, 625 = 5^8$
:The $25$th fourth power:
::$390 \, 625 = 25 \times 25 \times 25 \times 25$
:The $625$th square number:
::$390 \, 625 = 625 \times 625$
== Also see ==
* 
* 
* 
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Category:Specific Numbers
Category:390,625
391

== Number ==
$391$ (three hundred and ninety-one) is:
:$17 \times 23$
:The $17$th Smith number after $4$, $22$, $27$, $58$, $85$, $94$, $121$, $166$, $202$, $265$, $274$, $319$, $346$, $355$, $378$, $382$:
::$3 + 9 + 1 = 1 + 7 + 2 + 3 = 13$
== Also see ==
* 
Category:Specific Numbers
Category:391
3916

== Number ==
$3916$ (three thousand, nine hundred and sixteen) is:
:$2^2 \times 11 \times 89$
:The $88$th triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $3240$, $3321$, $3403$, $3486$, $3570$, $3655$, $3741$, $3828$:
::$3916 = \ds \sum_{k \mathop = 1}^{88} k = \dfrac {88 \times \paren {88 + 1} } 2$
== Also see ==
* 
Category:Specific Numbers
Category:3916
392

== Number ==
$392$ (three hundred and ninety-two) is:
:$2^3 \times 7^2$
:The $16$th second pentagonal number after $2$, $7$, $15$, $26$, $40$, $57$, $77$, $100$, $126$, $155$, $187$, $222$, $260$, $301$, $345$:
::$392 = \dfrac {16 \paren {3 \times 16 + 1} } 2$
:The $32$nd generalized pentagonal number after $1$, $2$, $5$, $7$, $12$, $15$, $\ldots$, $222$, $247$, $260$, $287$, $301$, $330$, $345$, $376$:
::$392 = \dfrac {16 \paren {3 \times 16 + 1} } 2$
:The $33$rd powerful number after $1$, $4$, $8$, $9$, $16$, $25$, $\ldots$, $144$, $169$, $196$, $200$, $216$, $225$, $243$, $256$, $288$, $289$, $324$, $343$, $361$
== Also see ==
* 
* 
* 
Category:Specific Numbers
Category:392
393

== Number ==
$393$ (three hundred and ninety-three) is:
:$3 \times 131$
:The $12$th palindromic lucky number:
::$1$, $3$, $7$, $9$, $33$, $99$, $111$, $141$, $151$, $171$, $303$, $393$, $\ldots$
== Also see ==
* 
Category:Specific Numbers
Category:393
394

== Number ==
$394$ (three hundred and ninety-four) is:
:$2 \times 197$
:The $3$rd candidate Lychrel number.
:The $40$th noncototient after $10$, $26$, $34$, $50$, $\ldots$, $344$, $346$, $362$, $366$, $372$, $386$:
::$\nexists m \in \Z_{>0}: m - \map \phi m = 394$
:where $\map \phi m$ denotes the Euler $\phi$ function
== Also see ==
* 
* 
Category:Specific Numbers
Category:394
3940

== Number ==
$3940$ (three thousand, nine hundred and forty) is:
:$2^2 \times 5 \times 197$
:The $40$th heptagonal number after $1$, $7$, $18$, $34$, $55$, $81$, $112$, $\ldots$, $2205$, $2356$, $2512$, $2673$, $2839$, $3010$, $3186$, $3367$, $3744$:
::$3940 = \ds \sum_{k \mathop = 1}^{40} \paren {5 k - 4} = \dfrac {40 \paren {5 \times 40 - 3} } 2$
== Also see ==
* 
Category:Specific Numbers
Category:3940
3943

== Number ==
$3943$ (three thousand, nine hundred and forty-three) is:
:The $547$th prime number
:The $18$th integer $n$ after $3$, $4$, $5$, $6$, $7$, $8$, $10$, $15$, $19$, $41$, $59$, $61$, $105$, $160$, $661$, $2653$, $3069$ such that $m = \ds \sum_{k \mathop = 0}^{n - 1} \paren {-1}^k \paren {n - k}!$ is prime
== Also see ==
* 
* 
Category:Specific Numbers
Category:3943
395

== Number ==
$395$ (three hundred and ninety-five) is:
:$5 \times 79$
:The $23$rd inconsummate number after $62$, $63$, $65$, $75$, $\ldots$, $371$, $372$, $377$, $381$, $383$, $386$, $387$:
::$\nexists n \in \Z_{>0}: n = 395 \times s_{10} \left({n}\right)$
== Sources ==
* 
Category:Specific Numbers
Category:395
396,733

== Number ==
$396 \, 733$ (three hundred and ninety-six thousand, seven hundred and thirty-three) is:
:The $33 \, 608$th prime number
:The smaller of the $1$st pair of primes whose prime gap is $100$:
::$396 \, 833 - 396 \, 733 = 100$
== Also see ==
* 
* 
== Sources ==
* : $396,733$
* : $396,733$
Category:Specific Numbers
Category:396,733
396,833

== Number ==
$396 \, 833$ (three hundred and ninety-six thousand, eight hundred and thirty-three) is:
:The $33 \, 609$th prime number
:The larger of the $1$st pair of primes whose prime gap is $100$:
::$396 \, 833 - 396 \, 733 = 100$
== Also see ==
* 
* 
== Sources ==
* : $396,733$
* : $396,733$
Category:Specific Numbers
Category:396,833
3969

== Number ==
$3969$ (three thousand, nine hundred and sixty-nine) is:
:$3^4 \times 7^2$
:The $63$rd square number after $1$, $4$, $9$, $16$, $25$, $36$, $\ldots$, $3249$, $3364$, $3481$, $3600$, $3721$, $3844$:
::$3969 = 63 \times 63$
== Also see ==
* 
Category:Specific Numbers
Category:3969
397

== Number ==
$397$ (three hundred and ninety-seven) is:
:The $78$th prime number
:The larger of the $3$rd pair of primes whose prime gap is $8$:
::$397 - 389 = 8$
:The $12$th centered hexagonal number after $1$, $7$, $19$, $37$, $61$, $91$, $127$, $169$, $217$, $271$, $331$:
::$397 = \ds 1 + \sum_{k \mathop = 1}^{12 - 1} 6 k = 12^3 - 11^3$
:The $31$st left-truncatable prime after $2$, $3$, $5$, $7$, $\ldots$, $197$, $223$, $283$, $313$, $317$, $337$, $347$, $353$, $367$, $373$, $383$
== Also see ==
* 
* 
* 
* 
Category:Specific Numbers
Category:397
399

== Number ==
$399$ (three hundred and ninety-nine) is:
:$3 \times 7 \times 19$
:The $1$st Lucas-Carmichael number:
::$\paren {3 + 1} \divides \paren {399 + 1}$, $\paren {7 + 1} \divides \paren {399 + 1}$, $\paren {19 + 1} \divides \paren {399 + 1}$
:The $5$th positive integer after $79$, $159$, $239$, $319$ which cannot be expressed as the sum of fewer than $19$ fourth powers:
::$399 = 14 \times 1^4 + 3 \times 2^4 + 3^4 + 4^4$
:or:
::$399 = 11 \times 1^4 + 4 \times 2^4 + 4 \times 3^4$
:The $15$th integer $m$ such that $m! + 1$ (its factorial plus $1$) is prime:
::$0$, $1$, $2$, $3$, $11$, $27$, $37$, $41$, $73$, $77$, $116$, $154$, $320$, $340$, $399$
:The $43$rd sphenic number after $30$, $42$, $66$, $70$, $\ldots$, $290$, $310$, $318$, $322$, $345$, $354$, $357$, $366$, $370$, $374$, $385$:
::$399 = 3 \times 7 \times 19$
== Also see ==
* 
* 
* 
* 
== Sources ==
* : $399$
* : $399$
Category:Specific Numbers
Category:399
399,877,410,625

== Number ==
$399 \, 877 \, 410 \, 625$ is:
:$5^4 \times 7 \times 11 \times 17 \times 23 \times 79 \times 269$
:The $26$th square pyramorphic number after $1$, $55$, $5525$, $\ldots$:
::$399 \, 877 \, 410 \, 625 = \ds \sum_{k \mathop = 1}^{10 \, 625} k^2 = \dfrac {10 \, 625 \paren {10 \, 625 + 1} \paren {2 \times 10 \, 625 + 1} } 6$
:The $10 \, 625$th square pyramidal number:
::$399 \, 877 \, 410 \, 625 = \ds \sum_{k \mathop = 1}^{10 \, 625} k^2 = \dfrac {10 \, 625 \paren {10 \, 625 + 1} \paren {2 \times 10 \, 625 + 1} } 6$
== Also see ==
* 
* 
== Sources ==
* : Chapter $37$: On the Existence of Cakemorphic Integers: Figure $37.4$: Square pyramorphic numbers
* : $399,877,410,625$
Category:Specific Numbers
Category:399,877,410,625
3997

== Number ==
$3997$ (three thousand, nine hundred and ninety-seven) is:
:$7 \times 571$
:The $37$th centered hexagonal number after $1$, $7$, $19$, $37$, $61$, $91$, $127$, $\ldots$, $2791$, $2977$, $3169$, $3367$, $3571$, $3781$:
::$3997 = \ds 1 + \sum_{k \mathop = 1}^{37 - 1} 6 k = 37^3 - 36^3$
== Also see ==
* 
Category:Specific Numbers
Category:3997
399 is not Expressible as Sum of Fewer than 19 Fourth Powers

== Theorem ==
$399$ cannot be expressed as the sum of fewer than $19$ fourth powers:
:$399 = 14 \times 1^4 + 3 \times 2^4 + 3^4 + 4^4$
or:
:$399 = 11 \times 1^4 + 4 \times 2^4 + 4 \times 3^4$
== Proof ==
First note that $5^4 = 625 > 399$.
Then note that $2 \times 4^4 = 512 > 399$.
Hence any expression of $399$ as fourth powers uses no $n^4$ for $n \ge 5$, and uses not more than $1$ instance of $4^4$.
For the remainder, using $2^4$ uses fewer fourth powers than $16$ instances of $1^4$ does.
Now we have:
and it can be seen that the first and the third uses the least number of fourth powers, at $19$.
== Also see ==
* Hilbert-Waring Theorem
* Smallest Number not Expressible as Sum of Fewer than 19 Fourth Powers
* 159 is not Expressible as Sum of Fewer than 19 Fourth Powers
* 319 is not Expressible as Sum of Fewer than 19 Fourth Powers
== Sources ==
* : $399$
* : $399$
Category:Fourth Powers
Category:Hilbert-Waring Theorem
Category:399
3^x + 4^y equals 5^z has Unique Solution

== Theorem ==
The Diophantine equation:
:$3^x + 4^y = 5^z$
has exactly one solution in (strictly) positive integers:
:$3^2 + 4^2 = 5^2$
== Proof ==
Rewriting our Diophantine equation modulo $4$ we have:
:$\paren {-1}^x + 0^y \equiv 1^z \pmod 4$
Therefore $x$ is even.
Rewriting our Diophantine equation modulo 3 we have:
:$0^x + 1^y \equiv \paren {-1}^z \pmod 3$
Therefore $z$ is even.
Since $x$ and $z$ must both be even, we will rewrite $x$ as $2 r$, $z$ as $2 s$ and notice that $4$ is $2^2$.
We now have:
We can see that the is a product consisting entirely of instances of $2$.
Therefore:
:$\paren {5^s + 3^r} = 2^u$
and:
:$\paren {5^s - 3^r} = 2^v$
where $u > v$ and $u + v = 2 y$.
Solving for $5^s$, we add the two equations and get:
Solving for $3^r$, we subtract the two equations and get:
Since both $5^s$ and $3^r$ are odd, $v$ must be equal to $1$.
Let $t = u - v$.
We now have:
:$5^s = \paren {2^t + 1}$
:$3^r = \paren {2^t - 1}$
Let us now consider the second equation Modulo 3:
:$0 \equiv \paren{-1}^t - 1 \pmod 3$
Therefore t must be even. 
We will rewrite the second equation above with $t$ as $2 w$:
We can see that the is a product consisting entirely of instances of $3$.
Therefore:
:$ 3^m = \paren {2^w + 1}$
and:
:$ 3^n = \paren {2^w - 1}$
where $m > n$ and $m + n = r$.
Subtracting the $2$ equations above we get:
Therefore $m = 1$ and $n = 0$.
From above:
:$3^m = \paren {2^w + 1}$
therefore:
:$w = 1$
Recall that $t = 2w$.
Therefore:
:$t = 2$
and:
:$5^s = \paren {2^t + 1}$
:$3^r = \paren {2^t - 1}$
Therefore $r = 1$ and $s = 1$.
Finally recall that $x = 2 r$ and $z = 2 s$.
Therefore we arrive at the only possible solution in (strictly) positive integers:
:$x = 2$, $y = 2$ and $z = 2$
== Sources ==
* 
* : $2$
Category:Number Theory
3^x + 4^y equals 5^z has Unique Solution/Mistake

== Source Work ==
:
:The Dictionary
::$2$
== Mistake ==
:Also, the only solution of $3^x + 4^y = 5^z$ in integers is $x = y = z = 2$. The equation $5^x + 12^y = 13^z$ has the same unique solution.
== Correction ==
What it should say is:
:Also, the only solution of $3^x + 4^y = 5^z$ in (strictly) positive integers is $x = y = z = 2$ ...
as we also have the solutions:
In the original work from which Wells's information was taken, the correct specification for $x$, $y$ and $z$ was used.
This factoid was not included in the first edition of .
== Sources ==
* : $2$
Category:Mistakes/Curious and Interesting Numbers
3 Configurations of 9 Lines with 3 Intersection Points on each Line

== Theorem ==
There exist exactly $3$ essentially different configurations of $9$ straight lines each of which has exactly $3$ points of intersection.
This is one: there are two others.
:
== Proof ==
== Also see ==
* Pappus's Hexagon Theorem
== Sources ==
* : $9$
* : $9$
Category:Projective Geometry
3 Numbers in A.P. whose 4th Powers are Sum of Four 4th Powers

== Theorem ==
The following triplets of integers in arithmetic sequence with common difference of $60$ can all be expressed as the sum of four $4$th powers:
:$\tuple {8373, 8433, 8493}, \tuple {8517, 8577, 8637}, \ldots$
== Proof ==
The internal structure of these numbers reveals an interesting pattern:
== Sources ==
* 
* : $8737$
* 
Category:Fourth Powers
3 Proper Integer Heronian Triangles whose Area and Perimeter are Equal

== Theorem ==
There are exactly $3$ proper integer Heronian triangles whose area and perimeter are equal.
These are the triangles whose sides are:
:$\tuple {6, 25, 29}$
:$\tuple {7, 15, 20}$
:$\tuple {9, 10, 17}$
== Proof ==
First, using Pythagoras's Theorem, we establish that these integer Heronian triangles are indeed proper:
Now we show they have area equal to perimeter.
We use Heron's Formula throughout:
:$\AA = \sqrt {s \paren {s - a} \paren {s - b} \paren {s - c} }$
where:
:$\AA$ denotes the area of the triangle
:$a$, $b$ and $c$ denote the lengths of the sides of the triangle
:$s = \dfrac {a + b + c} 2$ denotes the semiperimeter of the triangle.
Thus we take the $3$ triangles in turn:
It remains to be demonstrated that these are indeed the only such proper integer Heronian triangles which match the criterion.
Let $\tuple {a, b, c}$ be the sides of such a triangle.
Using Heron's Formula, we have:
Note that:
:$\paren {s - a} + \paren {s - b} + \paren {s - c} = 3 s - a - b - c = s$
Hence by substituting $x = s - a$, $y = s - b$, $z = s - c$:
:$4 \paren {x + y + z} = x y z$
By Semiperimeter of Integer Heronian Triangle is Composite, $s$ is an integer.
Hence $s, x, y, z \in \N_{>0}$.
By Triple with Product Quadruple the Sum, our equation has solutions:
:$\tuple {1, 5, 24}, \tuple {1, 6, 14}, \tuple {1, 8, 9}, \tuple {2, 3, 10}, \tuple {2, 4, 6}$
Using:
:$a = s - x = x + y + z - x = y + z$
:$b = s - y = x + z$
:$c = s - z = x + y$
the possible sets of side lengths are:
:$\tuple {29, 25, 6}, \tuple {20, 15, 7}, \tuple {17, 10, 9}, \tuple {13, 12, 5}, \tuple {10, 8, 6}$
of which the final $2$ are Pythagorean Triples, so they are not proper Heronian triangles.
Hence the result.
== Sources ==
* : Bachet: $110$
Category:Heronian Triangles
3 is Divisor of one of n, n+2, n+4

== Theorem ==
Let $n \in \Z$ be an integer such that $n > 3$.
Then at least one of $n$, $n + 2$, $n + 4$ is divisible by $3$.
== Proof ==
Let $n \in \Z_{>0}$.
Then $n$ can be represented as either:
Let $n = 3 k$.
Then $n$ is divisible by $3$.
Let $n = 3 k + 1$.
Then:
:$n + 2 = 3 k + 3 = 3 \paren {k + 1}$
which is divisible by $3$.
Let $n = 3 k + 2$.
Then:
:$n + 4 = 3 k + 6 = 3 \paren {k + 2}$
which is divisible by $3$.
Hence the result.
== Sources ==
* : Chapter $2$: Divisibility Theory in the Integers: $2.2$ The Greatest Common Divisor: Problems $2.2$: $4$
Category:Examples of Divisors of Integers
4,100,625

== Number ==
$4 \, 100 \, 625$ (four million, one hundred thousand, six hundred and twenty-five) is:
:$3^8 \times 5^4$
:The $4$th and last positive integer after $1$, $625$, $6561$ whose $4$th root equals the number of its divisors:
::$\map {\sigma_0} {4 \, 100 \, 625} = 45 = \sqrt [4] {4 \, 100 \, 625}$
:The $45$th fourth power:
::$4 \, 100 \, 625 = 45 \times 45 \times 45 \times 45$
:The $2025$th square number:
::$4 \, 100 \, 625 = 2025 \times 2025$
== Also see ==
* 
* 
* 
== Sources ==
* : $4,100,625$
Category:Specific Numbers
Category:4,100,625
4,194,304

== Number ==
$4 \, 194 \, 304$ (four million, one hundred and ninety-four thousand, three hundred and four) is:
:$2^{22}$
:The $11$th power of $4$ after $(1)$, $4$, $16$, $64$, $256$, $1024$, $4096$, $16 \, 384$, $65 \, 536$, $262 \, 144$, $1 \, 048 \, 576$:
::$4 \, 194 \, 304 = 4^{11}$
:The $22$nd power of $2$ after $(1)$, $2$, $4$, $8$, $16$, $\ldots$, $65 \, 536$, $131 \, 072$, $262 \, 144$, $524 \, 288$, $1 \, 048 \, 576$, $2 \, 097 \, 152$:
::$4 \, 194 \, 304 = 2^{22}$
:The $23$rd almost perfect number after $1$, $2$, $4$, $8$, $16$, $\ldots$, $65 \, 536$, $131 \, 072$, $262 \, 144$, $524 \, 288$, $1 \, 048 \, 576$, $2 \, 097 \, 152$:
::$\map {\sigma_1} {4 \, 194 \, 304} = 8 \, 388 \, 607 = 2 \times 4 \, 194 \, 304 - 1$
:The $2048$th square number:
::$4 \, 194 \, 304 = 2048 \times 2048$
== Also see ==
* 
* 
* 
* 
Category:Specific Numbers
Category:4,194,304
4,210,818

== Number ==
$4 \, 210 \, 818$ (four million, two hundred and ten thousand, eight hundred and eighteen) is:
:$2 \times 3 \times 19 \times 43 \times 859$
:The $22$nd pluperfect digital invariant after $1$, $2$, $3$, $4$, $5$, $6$, $\ldots$, $8208$, $9474$, $54 \, 748$, $92 \, 727$, $93 \, 084$, $548 \, 834$, $1 \, 741 \, 725$:
== Also see ==
* 
Category:Specific Numbers
Category:4,210,818
4,294,967,297

== Number ==
$4 \, 294 \, 967 \, 297$ is:
:$641 \times 6 \, 700 \, 417$
:The $6$th Fermat number after $3$, $5$, $17$, $257$, $65 \, 537$:
::$4 \, 294 \, 967 \, 297 = 2^{\paren {2^5} } + 1 = 2^{32} + 1$
:and the $1$st not to be prime.
== Also see ==
* Prime Decomposition of 5th Fermat Number
* 
== Sources ==
* : $257$
* : $4,294,967,297$
* : $257$
* : $4,294,967,297$
Category:Specific Numbers
Category:4,294,967,297