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Is it possible to find infinitely many numbers such that..?

• is it possible to find infinitely many natural numbers such that none of them can be written as sum of two square natural numbers and at the same time none of them can be written as sum of two cubic numbers..? I'm quite sure that, of course this is possible and to prove that we have to assume that there is a natural number n such that all natural numbers above n can be written even as a sum of square numbers even as a sum of cubic numbers..and after to prove that this is impossible.. but i don't know how to prove that.. if anybody think the opposite please correct me.. if something is not clear let me know..

• Answer:

  Any number of the form 9k+3 would neither be expressible as a sum of two squares nor of two cubes. Hence there are infinitely many such numbers. To show this, we just have to prove that the following congruences cannot hold: x² + y² = 3 mod 9 x³ + y³ = 3 mod 9 For x² + y² = 3 mod 9, we first consider what are the possible residues of squares mod 9. We only have to test from x = 0 to 4 because beyond that it's just -x and the square would remove the minus sign. So we have possible residues 0,1, 4, 7. So the possible residues for x² + y² would just be, taking all possible paired sums of {0,1,4,7}, 0,1,2,4,5,7,8, none of which are 3. Similarly for x³ + x³ = 3 mod 9, the possible residues of cubes mod 9 are 0,1,8. Possible residues for x³ + y³ would then be 0,1,2,7,8, none of which are 3. Remarks: [1] Although we didn't need this fact in the proof, note that none of the residues mod 9 can be 6 either. So numbers of the form 9k+6 would have worked as well. [2] The modulus 9 is the smallest possible. It was found by running a Python script.