Erdős–Moser equation

In number theory, the Erdős–Moser equation is

${\displaystyle 1^{k}+2^{k}+\cdots +m^{k}=(m+1)^{k}}$

where ${\displaystyle m}$ and ${\displaystyle k}$ are positive integers. The only known solution is 1¹ + 2¹ = 3¹, and Paul Erdős conjectured that no further solutions exist.

Constraints on solutions[]

Leo Moser in 1953 proved that 2 divides k and that there is no other solution with m < 10^1,000,000.

In 1966, it was shown that 6 ≤ k + 2 < m < 2k.

In 1994, it was shown that lcm(1,2,...,200) 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 × 10^9,321,155.

In 2002, it was shown that all primes between 200 and 1000 must divide k.

In 2009, it was shown that 2k / (2m – 3) must be a convergent of ln(2); large-scale computation of ln(2) was then used to show that m > 2.7139 × 10^{1,667,658,416}.

References[]