Lifting-the-exponent lemma

The lifting-the-exponent lemma computes the exponent of a prime dividing a difference or sum of powers, given suitable divisibility conditions. It reduces a valuation of a power expression to valuations involving its base and exponent.

p-adic valuation: The exponent of a prime in a nonzero integer, extended to rational numbers by subtraction. The lemma’s output is precisely a prime-adic valuation.
Congruence: An equivalence relation on integers expressing that their difference is divisible by a fixed modulus. LTE’s assumptions are commonly stated as congruences modulo a prime.
Olympiad number theory: The problem-solving branch of number theory featured in mathematical competitions. LTE often shortens competition problems about prime divisibility of large powers.
Hensel's lemma: A theorem that lifts solutions of polynomial congruences modulo prime powers under suitable derivative conditions. Both track behavior modulo prime powers, but Hensel’s lemma lifts roots rather than evaluating power-expression valuations.
Difference of powers: The identity expressing a difference of powers as a product, including xⁿ − yⁿ = (x − y)(xⁿ⁻¹ + ⋯ + yⁿ⁻¹). The lemma evaluates valuations of this expression by separating its factors.
Prime number: An integer greater than one whose only positive divisors are one and itself. The lemma concerns valuations at a prime, with different conditions for odd primes and 2.
Diophantine equation: An equation whose solutions are sought among integers or rational numbers. Valuation bounds from LTE can rule out integer solutions to power equations.
Binomial coefficients modulo a prime: The study of divisibility and congruences of binomial coefficients by primes. These congruences can support valuation arguments, but do not provide LTE’s power-difference formula.
LTE for odd primes: The standard lifting-the-exponent formula for an odd prime dividing x − y but not xy. It gives vₚ(xⁿ − yⁿ) = vₚ(x − y) + vₚ(n) under its hypotheses.
Divisibility: The relation in which one integer is a multiple of another integer. The lemma’s hypotheses specify which primes divide the bases and their differences or sums.

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