Pentagonal number theorem

Euler’s identity expands the infinite product ∏ₙ₌₁∞(1−qⁿ) as 1 plus alternating terms whose exponents are generalized pentagonal numbers.

Euler’s product identity: The identity ∏ₙ₌₁∞(1−qⁿ)=∑ₖ∈ℤ(−1)ᵏq^{k(3k−1)/2}. This is the theorem’s compact formula, with the bilateral sum exposing its exact sign pattern.
Partition function: The function p(n) counting the unordered decompositions of n into positive integers. Its values are recovered from the coefficients of the product’s reciprocal.
Leonhard Euler: An eighteenth-century Swiss mathematician whose work shaped analysis, number theory, and combinatorics. Euler published the identity and used it in studying integer partitions.
Partition congruence: A divisibility relation among partition numbers for integers in specified arithmetic progressions. The theorem’s product expansion supports identities used to establish partition congruences.
Euler’s pentagonal number recurrence: The recurrence for p(n) obtained by multiplying its generating function by Euler’s product identity. It is a coefficient-level consequence, rather than the product–series identity itself.
Generalized pentagonal number: An integer of the form k(3k−1)/2 for some integer k. These exponents determine every nonzero term in the theorem’s expansion.
Partition generating function: The series ∑ₙ≥0 p(n)qⁿ, equal to ∏ₙ≥1(1−qⁿ)⁻¹. Taking the reciprocal of the theorem’s product gives this standard encoding of partitions.
Introductio in analysin infinitorum: Euler’s 1748 textbook on infinite analysis and the foundations of function theory. Euler’s treatment of infinite products belongs to the mathematical work associated with this book.
Ramanujan congruences: Congruences for partition numbers, including p(5n+4)≡0 mod 5, p(7n+5)≡0 mod 7, and p(11n+6)≡0 mod 11. The theorem’s generating-function methods form part of the toolkit for partition congruences.
Binomial theorem: The theorem giving the finite expansion of (1+x)ⁿ for nonnegative integer n. Unlike finite binomial expansions, this identity relates an infinite product to an infinite sparse series.

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