Irrationality of factorial divisor-sum series
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For σₖ(n), the sum of kth powers of the positive divisors of n, is Σₙ≥1σₖ(n)/n! irrational for every integer k>0? The same formula at k=0 gives the divisor-count variant.
Why it matters and what counts as progress
Why it matters: Uniformity in k links an entire divisor-function family to irrationality rather than treating isolated factorial series.
What counts as progress: Prove irrationality uniformly over k or extend the method to other arithmetic functions and factorial-denominator series.
How the problem developed
- 1952 · Problem [1]
Paul Erdős Raised the k=1 irrationality case in an American Mathematical Monthly problem; the later solution established that case.
Kevin Pratt Proved the k=4 case, after earlier work had established the cases k=1, 2, and 3.
- 13 Sept 2026 · Formal result [3]
Tokengrinder Released a Lean development claiming irrationality of Σₙ σₖ(n)/n! for every natural k; the cases k≥1 cover Erdős Problem 252, and k=0 is an additional variant.
AI contribution
AI-assisted Lean proof resolves the factorial divisor-sum irrationality problem for all exponents
For every natural k≥0, the convergent series Σₙ≥1 σₖ(n)/n! is irrational, where σₖ(n) is the sum of kth powers of positive divisors of n. The k>0 cases answer Erdős Problem 252; the Lean theorem also handles k=0.
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