2009/10/07 by Chen, William Y. C., Lin, Bernard L. S.
#Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.0910.1263
We obtain congruences for the number a(n) of cubic partitions using modular forms. The notion of cubic partitions is introduced by Chan and named by Kim in connection with Ramanujan's cubic continued fractions. Chan has shown that a(n) has several analogous properties to the number p(n) of partitions, including the generating function, the continued fraction, and congruence relations. To be more specific, we show that a(25n+22)≡ 0 (\rm mod 5), a(49n+15) ≡ a(49n+29) ≡ a(49n+36) ≡ a(49n+43) ≡ 0 (\rm mod 7). Furthermore, we prove that a(n) takes infinitely many even values and infinitely odd values.