Sum of powers

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  2. Number theory
  3. Sum of powers

The following is a list of sum of powers: ap + bq = cr where gcd(a,b,c) = 1. If you have any comment please fill the form.

an + b3 = c2 (Finite number of solutions)

a3 + b2 = c7 (Finite number of solutions)

a3 + b2 = c9 (Finite number of solutions)

a3 + b3 = c2 (Infinite number of solutions)

a4 + b2 = c3 (Infinite number of solutions)

a4 + b3 = c2 (Infinite number of solutions)

a5 + b2 = c4 (Finite number of solutions)

a5 + b3 = c2 (Infinite number of solutions)

a5 + b4 = c2 (Finite number of solutions)

a7 + b3 = c2 (Finite number of solutions)

a8 + b2 = c3 (Finite number of solutions)

a8 + b3 = c2 (Finite number of solutions)

an + b3 = c2

  1. 1n + 23 = 32

a3 + b2 = c7

  1. 1 4143 + 2 213 4592 = 657
  2. 9 2623 + 15 312 2832 = 1137

a3 + b2 = c9

  1. 73 + 132 = 29

a3 + b3 = c2

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a4 + b2 = c3

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See 55 solutions with both a and b both less than 1012

a4 + b3 = c2

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See 1602 solutions with both a and b both less than 1012

a5 + b2 = c4

  1. 25 + 72 = 34

a5 + b3 = c2

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See 61 solutions with both a and b both less than 1012.

a5 + b4 = c2

  1. 35 + 114 = 1222

a7 + b3 = c2

  1. 27 + 173 = 712
  2. 177 + 76 2713 = 21 063 9282

a8 + b2 = c3

  1. 338 + 1 549 0342 = 15 6133

a8 + b3 = c2

  1. 438 + 96 2223 = 30 042 9072

Credits

The case x3 + y3 = z2 has been solved by Louis J. Mordell in his book "Diophantine equations", Academic Press, London-New York 1969.

Don Zagier found six parameterizations for x4 + y3 = z2. The last one was found by Johnny Edwards.

Some parametrizations of x5 + y3 = z2 are due to Don Zagier, Frits Beukers, and Steve Thiboutot. This is described in F.Beukers, Duke Math.J. 91 (1998), p61-88. A complete parameterization of this case was found by Johnny Edwards in 2001.

It was proved by Darmon and Granville in 1993 that if 1/p + 1/q + 1/r < 1 then the equation Axp + Byq + Czr = 0 has only finitely many solutions. See H. Darmon, A. Granville, On the equations zm = F(x,y) and Axp + Byq = Czr. Preprint 28 Volume II (1994), University of Georgia.

There are no other solutions for the exponent triple (2,3,8). It has been shown by Nils Bruin in Compositio Mathematicae 118 (1999) 305-321.

Last updated February 25th, 2011.