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bounding (y+1)^n-y^n, and a weaker FLT condition (Bulgaria 1967 P2)

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June 23, 2021
number theoryalgebrainequalities

Problem Statement

Prove that: (a) if y<12y<\frac12 and n≥3n\ge3 is a natural number then (y+1)n≥yn+(1+2y)n2(y+1)^n\ge y^n+(1+2y)^\frac n2; (b) if x,y,zx,y,z and n≥3n\ge3 are natural numbers for which x2−1≤2yx^2-1\le2y then xn+yn≠znx^n+y^n\ne z^n.
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