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I came across the Brahmagupta's identity mentioned here recently and thought it was pretty cool. https://en.wikipedia.org/wiki/Brahmagupta%27s_identity https://
by yantrams 1y ago
I came across the Brahmagupta's identity mentioned here recently and thought it was pretty cool. https://en.wikipedia.org/wiki/Brahmagupta%27s_identity https://en.wikipedia.org/wiki/Brahmagupta%27s_identity
It says - Numbers of the form a^2 + n*b^2 are closed under multiplication.
- deleted 1y ago[deleted]
- ogogmad 1y agoIt follows from matrices of the form [[a, (-n)b], [b, a]] being closed under multiplication, and taking determinants. In more advanced language: For R a commutative ring (like say, the integers) the following function f is a ring homomorphism f:R[√(-n)] -> M_2(R), f(a + b√(-n)) = [[a, (-n)b], [b, a]] Now take determinants.
- madcaptenor 1y agoThere's a nice characterization of sums of two squares in terms of their prime factorization), namely that all primes of form 4k+3 appear with even multiplicity. From a quick look through OEIS it looks like there are similar characterizations for a^2 + n*b^2 but this is where I tap out on number theory.