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Prove that two conguence statements are equivalent

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July 17, 2012
modular arithmeticinequalitiesnumber theory proposednumber theory

Problem Statement

Let 0<x<y<z<p0<x<y<z<p be integers where pp is a prime. Prove that the following statements are equivalent: (a)x3≡y3(modp) and x3≡z3(modp)(a) x^3\equiv y^3\pmod p\text{ and }x^3\equiv z^3\pmod p (b)y2≡zx(modp) and z2≡xy(modp)(b) y^2\equiv zx\pmod p\text{ and }z^2\equiv xy\pmod p
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