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 \(61,218,182,743,304,701,891,431,482,520\approx 6.12\times 10^{28}\).
• If odd perfect numbers exist, they must be at least as large as \(10^{1500}\).
• 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 \(2^{2^{2^{2^{2^{2^{2^{100n}}}}}}}\) 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.
• If they exist, counterexamples to the Collatz conjecture must be above \(2^{68}\)
• If there are only a finite number of twin primes, then the largest are at least 2996863034895 · 2^1290000-1 and 2996863034895 · 2^1290000+1.
• 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.