Numbers related to unsolved problems

These are large numbers that are related to unsolved problems. Some of these values are subject to change, e.g. bounds on solutions.
*If Singmaster's conjecture holds for \(N=6\), then the second smallest number that satisfies this condition in Pascal's triangle is \(61,218,182,743,304,701,891,431,482,520\approx 6.12\times 10^{28}\).T. D. Noe, Remark on sequence A003015 (2004)
*If odd perfect numbers exist, they must be at least as large as \(10^{1500}\).P. Ochem, M. Rao, Odd perfect numbers are greater than 10^1500 (2012, accessed 2020-11-11)
*A 2009 bound by Tao on the entries of sequences whose existence is proven by the Green-Tao theorem is that each entry is less than \(2^{2^{2^{2^{2^{2^{2^{100n}}}}}}}\)UCLA, Terence Tao: Structure and Randomness in the Prime Numbers (2009, accessed 2020-11-10) for a sequence of length \(n\).
*The cubes such that their sum isn't congruent to 4 or 5 mod 9 can be very large for small sums, for example \(42=(−80538738812075974)^3 + 80435758145817515^3 + 12602123297335631^3\).A. Sutherland, Sums of three cubes (2019) (p.24)
*If they exist, counterexamples to the must be above \(2^{68}\)D. Barina, Convergence Verification of the Collatz Problem (2020, accessed 2020-11-29)
*If there are only a finite number of twin primes, then the largest are at least 2996863034895×21290000-1 and 2996863034895×21290000+1.https://primes.utm.edu/primes/page.php?id=122213
*If the Goldbach conjecture is false, the smallest even number that cannot be written as a sum of two primes is at least 4·1018.http://sweet.ua.pt/tos/goldbach.html
*There are no known odd triperfect numbers below 1050NAJAR, R., and W. BECK. "LOWER BOUND FOR ODD TRIPERFECT NUMBERS." NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY. Vol. 22. No. 3. 201 CHARLES ST, PROVIDENCE, RI 02940-2213: AMER MATHEMATICAL SOC, 1975..
*If the fourth exists, it's above \(2\times 10^{13}\)Numberphile, What do 5, 13, and 563 have in common? (2013)
*The smallest known number of vertices in a counterexample to is \(\approx 4^{10,000}\)Numberphile, A Breakthrough in Graph Theory (2019) (accessed 2021-01-15)
*The endpoints of the have been verified up to \(4.5\times 10^{18}\)OEIS, Sequence A116086 (accessed 2021-01-20).
*If a exists, it must be above 1035Numberphile, Quasiperfect Numbers, (accessed 2021-01-22)
==Sources==
Category:Lists
Category:Unnamed numbers
Category:Numbers
Category:Unsolved problems
Category:Dynamic googolisms