On the Sum of the Reciprocals of the Fermat Numbers and Related Irrationalities

@article{Golomb1963OnTS,
  title={On the Sum of the Reciprocals of the Fermat Numbers and Related Irrationalities},
  author={Solomon W. Golomb},
  journal={Canadian Journal of Mathematics},
  year={1963},
  volume={15},
  pages={475 - 478}
}
  • S. Golomb
  • Published 1963
  • Mathematics
  • Canadian Journal of Mathematics
In (1), P. Erdös showed that the function takes on irrational values whenever z = 1/t, t = 2 , 3 , 4 , 5 , . . . . The method of proof uses Lambert's identity, where d(n) is the number of divisors of n; and it is shown that 
On the Irrationality of Certain Lucas Infinite Series
We consider the sequences of integers {un} defined by the recurrence relation un+1=pun-qun-1 (q≠0,q∈Z), and discuss the irrationality of ∑∞n=11un and ∑∞n=11vn in some conditions,
Addendum to: On the rational approximation of the sum of the reciprocals of the Fermat numbers
As a corollary of the main result of our recent paper, On the rational approximation of the sum of the reciprocals of the Fermat numbers published in this same journal, we prove that for each integer
Transcendence of a fast converging series of rational numbers
  • D. Duverney
  • Mathematics
    Mathematical Proceedings of the Cambridge Philosophical Society
  • 2001
Let a ∈ ℕ [setmn ] {0, 1} and let bn be a sequence of rational integers satisfying bn = O(η−2n) for every η ∈]0, 1[. We prove that the number S = [sum ]+∞n=0 1/(a2n + bn) is transcendental by using a
Fermat Numbers and Elite Primes
Fn+1 = (Fn − 1) + 1, n ≥ 0 and are pairwise coprime. It is conjectured that Fn are always square-free and that, beyond F4, they are never prime. The latter would imply that there are exactly 31
Irrationality of Fast Converging Series of Rational Numbers
We say that the series of general term un � 0 is fast converging if log |un |≤ c2 n for some c< 0. We prove irrationality results and compute irrationality measures for some fast converging se- ries
Some Diophantine Equations
This quotation from the preface of Mordell’s book, Diophantine Equations, Academic Press, London, 1969, indicates that in this section we shall have to be even more eclectic than elsewhere. If you’re
Some aspects of analytic number theory: parity, transcendence, and multiplicative functions
Questions on parities play a central role in analytic number theory. Properties of the partial sums of parities are intimate to both the prime number theorem and the Riemann hypothesis. This thesis
Jacobi continued fraction and Hankel determinants of the Thue-Morse sequence
Abstract We study the Jacobi continued fraction and the Hankel determinants of the Thue-Morse sequence and obtain several interesting properties. In particular, a formal power series φ(x) is being
...
...

References

On Arithmetical Properties of Lambert Series
rr=l u=l Chowla* has proved that if t is an integer 2 5, then g(x it) is irrational. He also conjectures that for rational 1 x I< J bothJ(x) and g(x) are irrational. In the present note we prove the