Proof of the Churchhouse conjecture concerning binary partitions

“…. , 2 k−1 } the ith summand has the same 2-adic valuation as the 0th one, or, in other words, we have equality in (9). Equality in (9) holds only if i = 2 k−1 or i = j − 2 k−1 .…”

Section: Lemma 32 Let M Be a Positive Integer Thenmentioning

confidence: 97%

“…In particular b 1 (n) = b(n) is the well known binary partition function introduced by Euler and studied by Churchhouse [5], Rødseth [16], Gupta [9] and others. It is sequence A018819 in [17].…”

Section: Arithmetic Properties Of the Sequence ( F (N T)) N∈n With Tmentioning

confidence: 99%

Arithmetic properties of coefficients of power series expansion of $$\prod _{n=0}^{\infty }\left( 1-x^{2^{n}}\right) ^{t}$$ ∏ n = 0 ∞

2017

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(1 − x 2 n ) be the generating function for the ProuhetThue-Morse sequence ((−1) s 2 (n) ) n∈N . In this paper we initiate the study of the arithmetic properties of coefficients of the power series expansions of the functionFor t ∈ N + the sequence ( f n (t)) n∈N is the Cauchy convolution of t copies of the Prouhet-Thue-Morse sequence. For t ∈ Z <0 the n-th term of the sequence ( f n (t)) n∈N counts the number of representations of the number n as a sum of powers of 2 where each summand can have one among −t colors. Among other things, we present a characterization of the solutions of the equations f n (2 k ) = 0, where k ∈ N, and f n (3) = 0. Next, we present the exact value of the 2-adic valuation of the number f n (1 − 2 m )-a result which generalizes the well known expression concerning the 2-adic valuation of the values of the binary partition function introduced by Euler and studied by Churchhouse and others.

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“…Within months, other mathematicians proved Churchhouse's conjectures and proved natural extensions of his results. These included Rødseth [8] who extended Churchhouse's results to include the functions b p (n) where p is any prime as well as Andrews [2] and Gupta [5,6] who proved that corresponding results also held for b m (n) where m could be any integer greater than 1. As part of an infinite family of results, these authors proved that, for any m ≥ 2 and any nonnegative integer n, b m (m(mn − 1)) ≡ 0 (mod m).…”

Section: Introductionmentioning

confidence: 99%