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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 $$61,218,182,743,304,701,891,431,482,520\approx 6.12\times 10^{28}$$.[1]
• If odd perfect numbers exist, they must be at least as large as $$10^{1500}$$.[2]
• 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 $$2^{2^{2^{2^{2^{2^{2^{100n}}}}}}}$$[3] 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 $$42=(−80538738812075974)^3 + 80435758145817515^3 + 12602123297335631^3$$.[4]
• If they exist, counterexamples to the Collatz conjecture must be above $$2^{68}$$[5]
• If there are only a finite number of twin primes, then the largest are at least 2996863034895×21290000-1 and 2996863034895×21290000+1.[6]
• 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.[7]
• There are no known odd triperfect numbers below 1050[8].
• If the fourth Wilson prime exists, it's above $$2\times 10^{13}$$[9]
• The smallest known number of vertices in a counterexample to Hedetniemi's conjecture is $$\approx 4^{10,000}$$[10]
• The endpoints of the Redmond-Sun conjecture have been verified up to $$4.5\times 10^{18}$$[11].
• If a quasiperfect number exists, it must be above 1035[12]

## Sources

1. T. D. Noe, Remark on sequence A003015 (2004)
2. P. Ochem, M. Rao, Odd perfect numbers are greater than 10^1500 (2012, accessed 2020-11-11)
3. UCLA, Terence Tao: Structure and Randomness in the Prime Numbers (2009, accessed 2020-11-10)
4. A. Sutherland, Sums of three cubes (2019) (p.24)
5. D. Barina, Convergence Verification of the Collatz Problem (2020, accessed 2020-11-29)
6. https://primes.utm.edu/primes/page.php?id=122213
7. http://sweet.ua.pt/tos/goldbach.html
8. NAJAR, 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.
9. Numberphile, What do 5, 13, and 563 have in common? (2013)
10. Numberphile, A Breakthrough in Graph Theory (2019) (accessed 2021-01-15)
11. OEIS, Sequence A116086 (accessed 2021-01-20)
12. Numberphile, Quasiperfect Numbers, (accessed 2021-01-22)