Ramanujan mentioned in his paper in 1920 that "it appears that there are no equally simple properties for any moduli involving primes other than these three"
$p(5n+4)\equiv0 \mod 5$
$p(7n+5)\equiv0 \mod 7$
$p(11n+6)\equiv0 \mod 11$
In 2003, Scott Ahlgren and Matthew Boylan published a paper "Arithmetic properties of the partition function", that confirmed the result.
I am attempting to understand the paper as part of my senior thesis for the next 1.5 years.
I would like to kindly ask:
- Why have elementary methods fail to prove the statement?
- What "machinery" do I need? I am under the impression that JP Serre's theory for p-adic modular forms will be key. Would anyone kindly recommend books for a systematic independent study on it?
- What would be the requisite knowledge required to understand the paper?
Thank you for your kind answers.