Explicit linear dependence congruence relations for the partition function modulo 4

Abstract Almost nothing is known about the parity of the partition function p ( n ), which is conjectured to be random. Despite this expectation, Ono [10] surprisingly proved the existence of infinitely many linear dependence congruence relations modulo 4 for p ( n ), indicating that the parity of the partition function cannot be truly random. Answering a question of Ono, we explicitly exhibit the first examples of these relations which he proved theoretically exist. The first two relations invoke 131 (resp. 198) different discriminants $$ D \le 24 k - 1 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>D</mml:mi> <mml:mo>≤</mml:mo> <mml:mn>24</mml:mn> <mml:mi>k</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> for $$ k = 309 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>=</mml:mo> <mml:mn>309</mml:mn> </mml:mrow> </mml:math> (resp. $$ k = 312 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>=</mml:mo> <mml:mn>312</mml:mn> </mml:mrow> </mml:math> ); new relations occur for $$ k = 316,\, 317,\, 319,\, 321,\, 322,\, 326, \, \ldots $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>=</mml:mo> <mml:mn>316</mml:mn> <mml:mo>,</mml:mo> <mml:mspace/> <mml:mn>317</mml:mn> <mml:mo>,</mml:mo> <mml:mspace/> <mml:mn>319</mml:mn> <mml:mo>,</mml:mo> <mml:mspace/> <mml:mn>321</mml:mn> <mml:mo>,</mml:mo> <mml:mspace/> <mml:mn>322</mml:mn> <mml:mo>,</mml:mo> <mml:mspace/> <mml:mn>326</mml:mn> <mml:mo>,</mml:mo> <mml:mspace/> <mml:mo>…</mml:mo> </mml:mrow> </mml:math> .