Erdős–Moser equation

In number theory, the Erdős–Moser equation is

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

Unsolved problem in mathematics:

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

(more unsolved problems in mathematics)

where $m$ and $k$ 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, in any further solutions, 2 must divide k and that m ≥ 10^1,000,000.[1]

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

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.[3]

Moser's method was extended in 1999 to show that m > 1.485 × 10^9,321,155.[4]

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}.[5]

References

Moser, Leo (1953). "On the Diophantine Equation 1^k + 2^k + ... + (m – 1)^k = m^k". Scripta Math. 19: 84–88.
Krzysztofek, B. (1966). "The Equation 1ⁿ + ... + mⁿ = (m + 1)ⁿ". Wyz. Szkol. Ped. W. Katowicech-Zeszyty Nauk. Sekc. Math. (in Polish). 5: 47–54.
Moree, Pieter; te Riele, Herman; Urbanowicz, J. (1994). "Divisibility Properties of Integers x, k Satisfying 1^k + 2^k + ... + (x – 1)^k = x^k". Math. Comp. 63: 799–815. Retrieved 2017-03-20.
Butske, W.; Jaje, L.M.; Mayernik, D.R. (1999). "The Equation Σ_p|N 1/p + 1/N = 1, Pseudoperfect Numbers, and Partially Weighted Graphs". Math. Comp. 69: 407–420. doi:10.1090/s0025-5718-99-01088-1. Retrieved 2017-03-20.
Gallot, Yves; Moree, Pieter; Zudilin, Wadim (2010). "The Erdős–Moser Equation 1^k + 2^k + ... + (m – 1)^k = m^k Revisited Using Continued Fractions". Mathematics of Computation. 80: 1221–1237. arXiv:0907.1356. doi:10.1090/S0025-5718-2010-02439-1. S2CID 16305654. Retrieved 2017-03-20.

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[1] Moser, Leo (1953). "On the Diophantine Equation 1^k + 2^k + ... + (m – 1)^k = m^k". Scripta Math. 19: 84–88.

[2] Krzysztofek, B. (1966). "The Equation 1ⁿ + ... + mⁿ = (m + 1)ⁿ". Wyz. Szkol. Ped. W. Katowicech-Zeszyty Nauk. Sekc. Math. (in Polish). 5: 47–54.

[3] Moree, Pieter; te Riele, Herman; Urbanowicz, J. (1994). "Divisibility Properties of Integers x, k Satisfying 1^k + 2^k + ... + (x – 1)^k = x^k". Math. Comp. 63: 799–815. Retrieved 2017-03-20.

[4] Butske, W.; Jaje, L.M.; Mayernik, D.R. (1999). "The Equation Σ_p|N 1/p + 1/N = 1, Pseudoperfect Numbers, and Partially Weighted Graphs". Math. Comp. 69: 407–420. doi:10.1090/s0025-5718-99-01088-1. Retrieved 2017-03-20.

[5] Gallot, Yves; Moree, Pieter; Zudilin, Wadim (2010). "The Erdős–Moser Equation 1^k + 2^k + ... + (m – 1)^k = m^k Revisited Using Continued Fractions". Mathematics of Computation. 80: 1221–1237. arXiv:0907.1356. doi:10.1090/S0025-5718-2010-02439-1. S2CID 16305654. Retrieved 2017-03-20.