Erdős–Moser equation

Unsolved problem in mathematics:

Does the Erdős–Moser equation have solutions other than $1^{1}+2^{1}=3^{1}$?

(more unsolved problems in mathematics)

In number theory, the Erdős–Moser equation is

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

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

Some care must be taken when comparing research on this equation, because some articles examining the problem study it in the form 1^k + 2^k + ... + m^k = (m + 1)^k instead.

Constraints on solutions

In 1953, Leo Moser proved that, in any further solutions, 2 must divide k and that m ≥ 10^1,000,000.

In 1966, it was shown that 6 ≤ k + 2 < m < 3 (k + 1) / 2.

In 1994, it was shown that lcm(1,2,...,200) divides k and that any prime factor of m must be irregular and > 10000.

In 1999, Moser's method was extended 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); in what the authors of that paper call “one of the very few instances where a large scale computation of a numerical constant has an application”, it was then determined that m > 2.7139 × 10^{1,667,658,416}.

In 2010, it was shown that

1. k is even,
2. m ≡ 3 (mod 8) and m ≡ ±1 (mod 3),
3. m – 1, (m + 1) / 2, 2m – 1, and 2m + 1 are all squarefree,
4. if p divides at least one of the numbers from the previous line, then p – 1 divides k, and
5. (m² – 1) (4m² – 1) / 12 is squarefree and has at least 4,990,906 prime factors.