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Dr Sardonicus · 2020-11-26, 00:48

There are algebraic formulas, e.g.

a = k*(p^2 - q^2), b = k*(2*p*q), c = k*(p^2 + q^2)

for which a^2 + b^2 = c^2 is a polynomial identity.

So k, p, and q can be rational integers, Gaussian integers, Eisenstein integers, arbitrary complex numbers, or just variables -- it's all good.

There is a book entitled Mathematics, Its Magic And Mastery by Aaron Bakst. One of its chapters is entitled Algebra, Boss of Arithmetic.

2020-11-26, 00:48	#3
Dr Sardonicus Feb 2017 Nowhere 10DC₁₆ Posts	There are algebraic formulas, e.g. a = k(p^2 - q^2), b = k(2pq), c = k(p^2 + q^2) for which a^2 + b^2 = c^2 is a polynomial identity. So k, p, and q can be rational integers, Gaussian integers, Eisenstein integers, arbitrary complex numbers, or just variables -- it's all good. There is a book entitled Mathematics, Its Magic And Mastery by Aaron Bakst. One of its chapters is entitled Algebra, Boss of Arithmetic*.