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Fermat's Divisor Problem

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Fermat's divisor problem, posed by Fermat in 1657, asks for solutions in positive integers x and y to

sigma(x^3)=y^2,

and to

sigma(x^2)=y^3,

where sigma(n) is the divisor function (Dickson 2005).

The first few solutions to sigma(x^3)=y^2 are (x,y)=(1,1), (7, 20), (751530, 1292054400), ... (OEIS A008849 and A048948). Lucas stated that there are an infinite number of solutions (Dickson 2005, p. 56), but only solutions up to the fourth are known to be complete.

The first few solutions to sigma(x^2)=y^3 are (x,y)=(1,1), (43098, 1729), ... (OEIS A008850 and A048949), with only solutions up to the second known to be complete.

See also

Divisor Function, Wallis's Problem

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References

Beiler, A. H. Recreations in the Theory of Numbers: The Queen of Mathematics Entertains. New York: Dover, p. 9, 1966.Dickson, L. E. History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Dover, pp. 54-58, 2005.Sloane, N. J. A. Sequences A008849, A008850, A048948, and A048949 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Fermat's Divisor Problem

Cite this as:

Weisstein, Eric W. "Fermat's Divisor Problem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/FermatsDivisorProblem.html

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