number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777� ...

, the Erdős–Moser equation is :

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

where

m

and

k

are positive integers. The only known solution is 1¹ + 2¹ = 3¹, and

Paul Erdős Paul Erdős ( hu, Erdős Pál ; 26 March 1913 – 20 September 1996) was a Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematical conjectures of the 20th century. pursued and proposed problems in ...

conjectured that no further solutions exist.

Constraints on solutions

Leo Moser Leo Moser (11 April 1921, Vienna – 9 February 1970, Edmonton) was an Austrian-Canadian mathematician, best known for his polygon notation. A native of Vienna, Leo Moser immigrated with his parents to Canada at the age of three. He received his ...

in 1953 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'' < 2''k''. 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 2''k'' / (2''m'' – 3) must be a convergent of

ln(2) The decimal value of the natural logarithm of 2 is approximately :\ln 2 \approx 0.693\,147\,180\,559\,945\,309\,417\,232\,121\,458. The logarithm of 2 in other bases is obtained with the formula :\log_b 2 = \frac. The common logarithm in particula ...

; large-scale computation of ln(2) was then used to show that ''m'' > 2.7139 × 10^{1,667,658,416}.

References

{{DEFAULTSORT:Erdős-Moser equation Diophantine equations Moser equation Unsolved problems in number theory