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单词 AnyNonzeroIntegerIsQuadraticResidue
释义

any nonzero integer is quadratic residue


Theorem.  For every nonzero integer a there exists an odd prime number p such that a is a quadratic residueMathworldPlanetmath modulo p.

Proof.1.a=2.  We see that  322(mod7)  and 72,  whence 2 is a quadratic residue modulo 7.
2.2a  but  a2.  The number  12-a=1-a  (which is odd and ±1) has an odd prime factor p which does not divide a.  Thus a is a quadratic residue modulo p.
3.a=3.  We state that  42-3=130(mod13)  and  133.  Therefore 3 is a quadratic residue modulo 13.
4.a=5.  We see that  42-5=110(mod11)  and  115, i.e. 5 is a quadratic residue modulo 11.
5.2a  but  a3,  a5.  Now the number  22-a=4-a  (which is odd and ±1) has an odd prime factor p.  Moreover, pa  since  p4.  Accordingly, a is a quadratic residue modulo p.

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更新时间:2025/5/4 5:01:00