Krizek’s Triangular-Square Divisor-Sum Characterization
Abstract
A positive triangular number is square exactly when its index and value have odd divisor sums.
Theorem 1.1 (Divisor-sum parity).
Proof. Machine-checked in Lean as D5/S3/Arith/KrizekTriangularSquareSigmaParity.sigma_odd_iff_square_or_twice_square (✓ std3). ∎
Source. Repository-derived.
Commentary.
For a positive natural number, the divisor sum is odd exactly when the number is a square or twice a square. The proof factors the divisor sum into prime-power geometric sums and characterizes the parity of each exponent at an odd prime. This criterion is also consumed by the twin-prime sigma-gcd results.
Theorem 1.2 (The triangular-square divisor-sum characterization).
Proof. Machine-checked in Lean as D5/S3/Arith/KrizekTriangularSquareSigmaParity.result (✓ std3). ∎
Resolves. Problems/oeis-a001108-krizek-triangular-square-sigma-parity (proved) by D5/S3/Arith/KrizekTriangularSquareSigmaParity.result.
Source. Repository-derived.
Acknowledgement. N. J. A. Sloane; Jaroslav Krizek (2016). OEIS A001108, a(n)-th triangular number is a square. URL: https://oeis.org/A001108.
Commentary.
The classical divisor-sum parity characterization reduces each odd divisor sum to a square-or-twice-square alternative. A coprime-product split for consecutive integers proves the forward direction, and a twice-square exclusion for triangular numbers proves the reverse direction. At n=0 the triangular number is zero and square, while its divisor sum is even, so positivity excludes that boundary.
References
- Truth anchor:
D5/S3/Arith/KrizekTriangularSquareSigmaParity.result - Truth anchor:
D5/S3/Arith/KrizekTriangularSquareSigmaParity.sigma_odd_iff_square_or_twice_square