Irrationality and Diophantine Approximation

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E0068/: Asks whether the sum over integers n at least two of one over n factorial minus one is irrational.

E0069/: Asks whether the sum over n of the number of distinct prime factors of n divided by two to the power n is irrational.

E0243/: Asks whether a sequence whose terms are asymptotically the square of the previous one, with rational reciprocal sum, must eventually satisfy a fixed recurrence.

E0247/: Asks whether the sum of two to the power minus a-n is transcendental whenever the increasing integer sequence a-n has unbounded ratio to n.

E0249/: Asks whether the sum over n of Euler's totient function of n divided by two to the n is irrational.

E0250/: Asks whether the sum over n of the sum-of-divisors function of n divided by two to the n is irrational.

E0251/: Asks whether the sum over n of the nth prime divided by two to the n is irrational.

E0252/: Asks whether, for every positive k, the sum over n of the sum of kth powers of the divisors of n divided by n factorial is irrational; answered yes by a 2026 kernel-checked Lean proof rebuilt, replayed and statement-audited here.

E0257/: Asks whether the sum of one over two to the n minus one, taken over any infinite set of naturals, is irrational.

E0258/: States that the divisor-function product-denominator series is irrational for every positive-integer sequence tending to infinity.

E0259/: Asks whether the sum over squarefree n of n divided by two to the n is irrational.

E0260/: Asks whether the sum of a-n divided by two to the a-n is irrational for every increasing sequence whose ratio to n tends to infinity.

E0262/: Asks how slowly an increasing integer sequence can grow while every sum of reciprocals of positive integer multiples of its terms stays irrational.

E0263/: Asks whether two to the two to the n keeps reciprocal sums irrational under all asymptotically equal replacements, and whether such sequences must grow fast.

E0264/: The powers-of-two case is false; asks whether factorial denominators keep their reciprocal sum irrational under every bounded nonzero integer perturbation.

E0265/: Asks how fast an increasing integer sequence can grow when the sum of reciprocals of its terms and of its terms minus one are both rational.

E0266/: Asks whether every sequence of positive integers with convergent reciprocal sum admits a positive integer shift making the shifted reciprocal sum irrational.

E0267/: Asks whether the sum of reciprocals of Fibonacci numbers along any geometrically growing index sequence must be irrational.

E0269/: Asks whether the sum of reciprocals of the least common multiples of the first n integers built from a fixed finite set of primes is irrational.

E0270/: Asks whether the sum over n of one over the product of the f of n consecutive integers starting at n plus one is irrational whenever f tends to infinity.

E0495/: Asks whether, for all pairs of real numbers, the product of n with the distances from n times each number to the nearest integer has limit inferior zero.

E0496/: Asks whether, for every positive irrational alpha, sums of two positive squares come arbitrarily close to alpha times a positive square; the site's wording over every irrational alpha fails at negative ones.

E0998/: Asks whether an interval with bounded discrepancy along the fractional multiples of an irrational alpha must have length a fractional multiple of alpha; the site's wording, asking that of both endpoints, is false.

E0999/: Asks whether almost-all approximability of alpha by fractions within f of q over q equals divergence of the sum of the totient of q times f of q over q.

E1000/: Concerns how many fractions with denominator the kth term of an integer sequence reduce to a denominator not equal to an earlier term of the sequence.

E1001/: Asks whether the measure of reals well approximated by fractions with denominator between N and cN tends to a limit, and what that limit is explicitly.

E1002/: Asks whether the logarithmically normalized sum of one half minus fractional parts of multiples of a fixed irrational has an asymptotic distribution function.

E1049/: Asks whether the sum over all n of one over t to the power n minus one is irrational for every rational t greater than one.

E1050/: Asks whether the sum over all n of one over two to the power n minus three is irrational.

E1051/: Asks whether the sum of one over consecutive products of an increasing integer sequence is irrational whenever the sequence grows doubly exponentially.

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Irrationality and transcendence of series and constants, irrationality sequences, and how well real numbers can be approximated by rationals.

Site tags routed here: analysis, diophantine approximation, irrationality, number theory.