“Erdős–Moser equation”的意思、由来-开放百科全书

词条

Erdős–Moser equation

释义

Constraints on solutions
References

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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

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 – 1) 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

{{cite journal

|last=Gallot
|first=Yves
|last2=Moree
|first2=Pieter
|last3=Zudilin
|first3=Wadim
|author-link=Yves Gallot
|author-link2=Pieter Moree
|author-link3=Wadim Zudilin
|year=2010
|title=The Erdős–Moser Equation 1^k + 2^k + ... + (m – 1)^k = m^k Revisited Using Continued Fractions
|journal=Mathematics of Computation
|volume=80
|pages=1221–1237
|language=English
|ref=harv
|url=https://www.ams.org/journals/mcom/2011-80-274/S0025-5718-2010-02439-1
|accessdate=2017-03-20}}

{{cite journal

|last=Moser
|first=Leo
|author-link=Leo Moser
|year=1953
|title=On the Diophantine Equation 1^k + 2^k + ... + (m – 1)^k = m^k
|journal=Scripta Math.
|volume=19
|pages=84–88
|language=English
|ref=harv}}

{{cite journal

|last=Butske
|first=W.
|last2=Jaje
|first2=L.M.
|last3=Mayernik
|first3=D.R.
|year=1999
|title=The Equation Σ_p|N 1/p + 1/N = 1, Pseudoperfect Numbers, and Partially Weighted Graphs
|journal=Math. Comp.
|volume=69
|pages=407–420
|language=English
|url=https://www.ams.org/journals/mcom/2000-69-229/S0025-5718-99-01088-1/
|accessdate=2017-03-20
|ref=harv}}

{{cite journal

|last=Krzysztofek
|first=B.
|year=1966
|title=The Equation 1ⁿ + ... + mⁿ = (m + 1)ⁿ
|journal=Wyz. Szkol. Ped. w. Katowicech-Zeszyty Nauk. Sekc. Math.
|volume=5
|pages=47–54
|language=Polish
|ref=harv}}

{{cite journal

|last=Moree
|first=Pieter
|last2=te Riele
|first2=Herman
|last3=Urbanowicz
|first3=J.
|author-link=Pieter Moree
|author-link2=Herman te Riele
|year=1994
|title=Divisibility Properties of Integers x, k Satisfying 1^k + 2^k + ... + (x – 1)^k = x^k
|journal=Math. Comp.
|volume=63
|pages=799–815
|language=English
|url=https://www.ams.org/journals/mcom/1994-63-208/S0025-5718-1994-1257577-1/
|accessdate=2017-03-20
|ref=harv}}{{DEFAULTSORT:Erdős-Moser equation}}

4 : Number theory|Diophantine equations|Paul Erdős|Unsolved problems in mathematics

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