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made the following important conjecture: For every positive integer m, on
every arithmetic progression r (mod m); 0 ≤ r < m – 1, p(n) assumes both even and odd values infinitely often. For the cases m = 2, 3, 4, 5, 6, 8, 10, 12, 16, 20, 40, the conjecture had
been established by the efforts of many mathematicians. In fact, the works of
Subbarao, Hirschhorn, and
Subbarao and Hirschhorn
among others like Frank Garvan and D. Stanton
established these cases by elegant combinatorial methods. This conjecture generated a lot of research by others in the field. A major achievement toward proving this
conjecture was made by K. Ono, whose result is as follows: following result
of K. For every positive integer m, in every arithmetic progression r (mod m), 0 ≤ r < m – 1, p(n) assumes even values infinitely often. Also, if p(n) assumes odd value for a single n in an arithmetical progression, then p(n) assumes odd values for infinitely many n in that arithmetical progression. Quantitative versions of the above result were obtained later by J. P. Serre and S. Ahlgren. The odd
case of the conjecture is still open, but has been verified for all m ≤
10^{5}. Dr. Subbarao offered a $500 prize for a complete proof of his
conjecture. He had an analogous conjecture for product partitions. Many other conjectures and unsolved problems appear in his papers with Erdös, Strauss, Katai, Hardy et al. Here is another example: If p_{1}, . . . ,
p_{r} are any distinct primes and a_{1}, . . . , a_{r} positive integers, then ∏ (p_{i}^{ai}  1) divides (( ∏ p_{i}^{ai}) – 1) only if r = 1. In one of his papers in 1966, Dr. Subbarao investigated the integer valued additive functions, and characterized the power function (with positive integer exponent among them) with some congruence condition, and formulated some interesting conjectures. This paper also inspired extensive research, the results of which were formulated in more than 20 research papers. Formulating open research problems was a typical character of his research
activity. By this means, he successfully
activated the research of other mathematicians, and thus contributed to the
growth of the field as a whole. 

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