[seqfan] Re: A107008 = primes of the form 24k+1?

Peter Munn techsubs at pearceneptune.co.uk
Mon Jun 8 20:30:14 CEST 2020


Following a private response on this subject, I should emphasize that the
tricky part seems to be to show that _all_ primes congruent to 1 mod 24
are in A107008. And after a little more spreadsheet work, it is starting
to look particularly interesting...

Does anyone want to check to what extent the following hypothesis is true?:

When k can be written as the sum of a square and a generalized pentagonal
number in exactly one way, the resulting numbers of the form 24k+1 might
be exactly the prime numbers of the form 24k+1 plus the squares of prime
numbers congruent to 13, 17, 19 or 23 mod 24.

Peter

On Mon, June 8, 2020 5:01 pm, Peter Munn wrote:
> Hi seqfans.
> In A107008 (Primes of the form x^2+24*y^2), NJAS comments "Presumably
this
> is the same as primes congruent to 1 mod 24."
> Can we come up with something to decide this?
> It would help to establish when 24k+1 can be written as x^2+24*y^2. I
reckon this happens when the k in 24k+1 can be written as the sum of a
square, j, and a generalized pentagonal number, i, because setting y =
sqrt(j), x = sqrt(24i+1) can be shown to satisfy the equation. (See Zak
Seidov's 2008 comment in http://oeis.org/A001318, "Generalized
pentagonal
> numbers".) I believe the converse is true, also.
> OEIS does not yet have the sequence "Numbers that can't be written as
the
> sum of a square and a generalized pentagonal number" [1], but the
sequence, S, starts 20, 29, 33, 34, 45, 46, 53, ... , and I reckon it
has
> positive asymptotic density.
> So the question becomes: if k is a term of S, why should 24k+1 be
composite, at least up to the limit of Vladimir Orlovsky's check in
A107008? From the first 3 terms we get: 20*24 + 1 = 481 = 13*37; 29*24 +
1
> = 697 = 17*41; 33*24 + 1 = 793 = 13*61.
> Going through more terms, I saw a pattern emerge, prompting me to ask:
is
> this particular subset of "24k+1" numbers the same as "nonsquare numbers
of the form (24i + m) * (24j + m), 0 <= i < j, m in {13, 17, 19, 23}"?
This would be interesting anyway, and could be a clue.
> However, I'm not sure I'm close to an answer, and there might be a much
easier route: does anyone have better ideas? Or know the answer already?
Best regards,
> Peter
> [1] I also tried looking for the number of ways positive integers _can_
be
> so written, using "Search: seq:2,2,1,1,2,2,1,1,2,1,2,1,1,1,1,4" but it
draws a blank.
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