# New parity results of sums of partitions and squares in arithmetic progressions

## DOI:

https://doi.org/10.11575/cdm.v14i1.62644## Keywords:

Number Theory## Abstract

Recently, Ballantine and Merca proved that if $ (a,b) \in \{(6,8),\ (8,12),\ (12,24),\ (15,40),\\ (16,48),\ (20,120),\ (21,168)\}$, then $\sum_{ak+1 \ {\rm square}}p(n-k)\equiv 1\ ({\rm mod}\ 2)$ if and only if $bn+1$ is a square. In this paper, we investigate

septuple $(a_1,a_2,a_3,a_4,a_5,a_6,a_7)\in \mathbb{N}^5\times \mathbb{Q}^2$ for which $\sum_{a_1k+a_2 \ {\rm square}}p(a_3a_4^\alpha n+a_6 a_4^\alpha+a_7-k)

\equiv 1\ ({\rm mod}\ 2)$ if and only if $a_5n+1$ is a square.

We prove some new parity results of sums of partitions and squares in arithmetic progressions which are analogous to the results due to Ballantine and Merca.

## References

partitions for $l\in\{5, 6, 7, 49\}$,

Ramanujan J. 40 (2014) 649--668.

-----------------------------------------------------------

C. Ballantine and M. Merca,

Parity of sums of partition numbers and squares

in arithmetic progressions, Ramanujan J., to appear (DOI:

10.1007/s11139-016-9845-6).

-----------------------------------------------------------

B.C. Berndt, Ramanujan's Notebooks, Part III,

Springer, New York,

1991.

-----------------------------------------------------------

N. Calkin, N. Drake, K. James, S. Law, P. Lee, D. Penniston and J.

Radder, Divisibility properties of the $5$-regular and $13$-regular

partition functions, Integers 8 (2008) \#A60

(see:http://www.ces.clemson.edu/~janoski/reu/2010/FiveRegularParity-2008-04-29.pdf).

-----------------------------------------------------------

M.D. Hirschhorn, On the residue mod 2 and 4 of $p(n)$,

Acta Arith.

38 (1980) 105--109.

-----------------------------------------------------------

M.D. Hirschhorn, On the parity of $p(n)$ II,

J. Combin.

Theory Ser. (A) 62 (1993)

128--138.

-----------------------------------------------------------

M.D. Hirschhorn and J.A. Sellers,

Elementary proofs of parity

results for 5-regular partitions,

Bull. Aust. Math. Soc. 81 (2010)

58--63.

-----------------------------------------------------------

O. Kolberg, Note on the parity of the partition function, Math.

Scand. 7 (1959) 377--378.

-----------------------------------------------------------

M. Newman, Periodicity modulo $m$ and divisibility properties of the

partition function, Trans. Amer. Math. Soc. 97 (1960) 225--236.

-----------------------------------------------------------

J.L. Nicolas, I.Z. Ruzsaand A. S\'{a}rk\"{o}zy,

On the parity of

additive representation functions (appendix by J.-P. Serre), J.

Number Theory 73 (1998) 292--317.

-----------------------------------------------------------

K. Ono, Parity of the partition function in

arithmetic progressions,

J. Reine Angew. Math. 472 (1996) 1--15.

-----------------------------------------------------------

K. Ono, Odd values of the partition function, Discrete Math. 169

(1997) 263--268.

-----------------------------------------------------------

T.R. Parkin and D. Shanks,

On the distribution of parity in the

partition function, Math. Comp. 21 (1967) 466--480.

-----------------------------------------------------------

C.S. Radu, A proof of Subbarao's conjecture,

J. Reine Angew. Math.

672 (2012) 161--175.

-----------------------------------------------------------

M.V. Subbarao,

Some remarks on the partition function, Amer. Math.

Month. 73 (1966) 851--854.

-----------------------------------------------------------

E.X.W. Xia,

Congruences modulo 9 and 27 for overpartitions,

Ramanujan J. 42 (2017) 301--323.

-----------------------------------------------------------

E.X.W. Xia and O.X.M. Yao, Analogues of Ramanujan's partition

identities, Ramanujan J. 31 (2013) 373--396.

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