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Pythagorean triples exist for natural numbers, but other exotic number sets lack them. For example, integers modulo 3: the square of both 1 and 2 is 1, so the
by HelloNurse 3y ago
Pythagorean triples exist for natural numbers, but other exotic number sets lack them.
For example, integers modulo 3: the square of both 1 and 2 is 1, so the sum of two nonzero squares is always 1 + 1 = 2, which is not a square.
Looking for infinite number sets without Pythagorean triples should be fun.
- Someone 3y agoOn the positive side, in integers modulo 3, we have n³ = n so 1³ + 2³ = 0³
- HelloNurse 3y agoPythagorean triples must be three nonzero numbers, and the exponent must be 2. If you are interested in other equations, don't call them the same way; and more importantly, while generalizing to other exponents and sums of more than two numbers is reasonable, allowing 0 makes the problem trivial because all powers of 0 are 0 and the sum of any number of zeros is 0.
- Someone 3y ago> If you are interested in other equations, don't call them the same way; I didn’t do that. I implicitly alluded to Fermat’s last theorem, expecting readers to pick that up. > because allowing 0 makes the problem trivial because all powers of 0 are 0 The example I gave isn’t trivial. The trivial ones are x³ + 0³ = x³ Also, nitpicking, not all powers of 0 are 0. The convention is that 0⁰ = 1 (https://en.wikipedia.org/wiki/Zero_to_the_power_of_zero https://en.wikipedia.org/wiki/Zero_to_the_power_of_zero)