Erdős–Moser equation


In number theory, the Erdős–Moser equation is
where and are positive integers. The only known solution is 11 + 21 = 31, and Paul Erdős conjectured that no further solutions exist.

Constraints on solutions

in 1953 proved that 2 divides k and that there is no other solution with m < 101,000,000.
In 1966 it was shown that 6 ≤ k + 2 < m < 2k.
In 1994 it was shown that lcm divides k and that any prime factor of m + 1 must be irregular and > 10000.
Moser's method was extended in 1999 to show that m > 1.485 × 109,321,155.
In 2002 it was shown that all primes between 200 and 1000 must divide k.
In 2009 it was shown that 2k / must be a convergent of ln; large-scale computation of ln was then used to show that m > 2.7139 × 101,667,658,416.