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Is 13477627276039606281933936961 the sum of two fourth powers? I suspect that this number is the sum of two fourth powers. Can anyone use Wolfram Mathematica or SAGE to check whether this number is the sum of two fourth powers ?

Derak
  • 547

1 Answers1

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The number is small enough to brute force.

Edit: fixed number

import gmpy2
N = 13477627276039606281933936961

for x in range(1, int((N**0.25)/2)+1):
    if gmpy2.is_square(N - x**4):
        print(x, (N-x**4)**0.25)

Output: 

3330955 10749966.0

>>> 3330955**4 + 10749966**4 == 13477627276039606281933936961
True
qwr
  • 11,362
  • Thanks very much qwr !! What software package or programming language did you use to solve this problem ? I use wolfram Mathematica, but the following command Solve[x^4+y^4 == 13477627276039606281933936961 , {x,y} , Integers] fails to solve this problem. Mathematica claims that there are no integers x and y which satisfy this equation. But clearly as you have shown , there are !! – Derak Apr 30 '17 at 19:36
  • @Derek I tried Mathematica as well, but it did not actually solve the problem. I used python here, which is not the fastest language but for $N^{1/4} \approx 10^7$ it is good enough. – qwr Apr 30 '17 at 19:41
  • @qwr In this post a few months from this one, the OP mentions that he was looking for counter-examples to Euler's Quartic Conjecture (already settled by Elkies). A database of $a^4+b^4+c^4=d^4$ can be found here but the terms $3330955^4 + 10749966^4$ do not appear. It's unfortunate the OP didn't clarify what this sum of two fourth powers would imply. (OP is no longer active in MSE.) – Tito Piezas III Sep 25 '23 at 16:58