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Proof of the Churchhouse conjecture concerning binary partitions

Published online by Cambridge University Press:  24 October 2008

Hansraj Gupta
Hansraj Gupta
Affiliation:
Allahabad University, Allahabad, India
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Extract

In a recent paper, R. F. Churchhouse has studied the function b(n) giving the number of partitions of n into powers of 2. The generating function of b(n) is

so that

Using this relation, Churchhouse has shown that

and

where

with

Moreover

and b(n) is even for n ≥ 2, while b(2n) ≡ (mod 4), for n = 22m−l(2k + 1); and, it is not ≡ (mod 8) for any n. He has conjectured that for k ≥ 1 and n odd,

and

The object of this note is to prove these conjectures.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 70 , Issue 1 , July 1971 , pp. 53 - 56
Copyright
Copyright © Cambridge Philosophical Society 1971

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References

REFERENCE

Crurcrhouse, R. F.Proc. Cambridge Philos. Soc. 66 (1969), 371–376.Google Scholar