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Modular Proof Help!!?
Prove that a number 10^3n+1, where n is a positive integer, cannot be represented as the sum of two cubes of positive integers.
Show me your working using modular theorems
10^(3n+1)
1 Answer
- coolkid70Lv 41 decade agoFavorite Answer
Hello.
Look at the cubes modulo 7. They can be 0, 1, 6 mod 7. But 10^{3n + 1} attains the values 3 or 4 mod 7. You can see that no sum of two cubes modulo 7 can equal either 3 or 4. For example, 0 + 0 = 0 mod 7; 0 + 1 = 1 mod 7; 0 + 6 = 6 mod 7; 1 + 1 = 2 mod 7; etc... (and you see that no combination will sum to 3 or 4).
This shows that no number of the form 10^{3n + 1} can be written as the sum of two cubes.
Good luck.
