Erdős–Moser Equation Proof (Part 11 - Setting the Boundaries and Definition)

In number theory, the Erdős–Moser equation is defined as

1k+2k+...+mk=(m+1)k1^k + 2^k + ... + m^k = (m + 1)^k

Of course, a well-known trivial solution is

11+21=311^1 + 2^1 = 3^1

However, Paul Erdős conjectured that no further solutions exist to the Erdős–Moser equation other than

11+21=311^1 + 2^1 = 3^1

So, now what?

Well, a solution to find mm is

mk(m+1)k=12km^k - (m + 1)^k = - 1 - 2^k

since

1k=11^k = 1

Leo Moser also posed certain constraints on the solutions:

11. 2k=x\frac{2}{k} = x, where xx is a positive number and that there is no other solution for m<101,000,000m < 10^{1,000,000}, which Leo Moser himself proved in 19531953.

22. k+2<m<2kk + 2 < m < 2k, which was shown in 19661966.

33. lcm(1,2,...,200)lcm (1, 2, ..., 200) divides kk and that any prime factor of m+1m + 1 must be irregular and >10,000> 10,000, which was shown in 19941994.

44. In 19991999, Moser's method was extended to show that m>1,485×109,321,155m > 1,485 \times 10^{9,321,155} .

55. In 20022002, it was shown that 200<p<1000200 < p < 1000 must divide kk, where pp is all the primes in between.

66. In 20092009, it was shown that 2k2m3\frac{2k}{2m - 3} must be a convergent of In(2)In (2) - large-scale computation was used to show that m>2.7139×101,667,658,416m > 2.7139 \times 10^{1,667,658,416}.

Note by Yajat Shamji
5 months, 3 weeks ago

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