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Re: square root modulo composite number distribution

2009-11-06T20:16:26Z

Actually Hensel's Lemma (Newton iteration) isn't needed since there's
a simple formula to lift Z/pZ sqrts to Z/(p^e)Z [due to Tonelli iirc]
LEMMA yy = a (mod p) => xx = a (mod p^e)
for x = y^c a^d, c = p^(e-1), d = ((p-2)c+1)/2
PROOF xx = a^c a^((p-2)c+1)
= a^((p-1)c) a
= a via (p-1)c = phi(p^e)

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Re: square root modulo composite number distribution

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Not always. x^2 = 1 (mod 2) has one and only one solution modulo 2.
And x^2=0 (mod p) has one and only one solution modulo p for each
prime p.
*But* if p>2 and a=/= 0 (mod p), then x^2=a (mod p) has either no
solutions or two solutions.
Sometimes.
Suppose that n is an odd composite number, and a is an integer, 0 < a