Erdős–Moser equation
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Unsolved problem in mathematics: Does the Erdős–Moser equation have solutions other than [math]\displaystyle{ 1^1+2^1=3^1 }[/math]? (more unsolved problems in mathematics)
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In number theory, the Erdős–Moser equation is
- [math]\displaystyle{ 1^k+2^k+\cdots+m^k=(m+1)^k, }[/math]
where [math]\displaystyle{ m }[/math] and [math]\displaystyle{ k }[/math] 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 ≥ 101,000,000.[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 × 109,321,155.[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 × 101,667,658,416.[5]
References
- ↑ Moser, Leo (1953). "On the Diophantine Equation 1k + 2k + ... + (m – 1)k = mk" (in English). Scripta Math. 19: 84–88.
- ↑ Krzysztofek, B. (1966). "The Equation 1n + ... + mn = (m + 1)n" (in Polish). Wyz. Szkol. Ped. W. Katowicech-Zeszyty Nauk. Sekc. Math. 5: 47–54.
- ↑ Moree, Pieter; te Riele, Herman; Urbanowicz, J. (1994). "Divisibility Properties of Integers x, k Satisfying 1k + 2k + ... + (x – 1)k = xk" (in English). Math. Comp. 63: 799–815. https://www.ams.org/journals/mcom/1994-63-208/S0025-5718-1994-1257577-1/. Retrieved 2017-03-20.
- ↑ Butske, W.; Jaje, L.M.; Mayernik, D.R. (1999). "The Equation Σp|N 1/p + 1/N = 1, Pseudoperfect Numbers, and Partially Weighted Graphs" (in English). Math. Comp. 69: 407–420. doi:10.1090/s0025-5718-99-01088-1. https://www.ams.org/journals/mcom/2000-69-229/S0025-5718-99-01088-1/. Retrieved 2017-03-20.
- ↑ Gallot, Yves; Moree, Pieter; Zudilin, Wadim (2010). "The Erdős–Moser Equation 1k + 2k + ... + (m – 1)k = mk Revisited Using Continued Fractions" (in English). Mathematics of Computation 80: 1221–1237. doi:10.1090/S0025-5718-2010-02439-1. https://www.ams.org/journals/mcom/2011-80-274/S0025-5718-2010-02439-1. Retrieved 2017-03-20.
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