Re: Number Theory Conjecture TOMMYS CONJECTURE
- From: quasi <quasi@xxxxxxxx>
- Date: Tue, 11 Sep 2007 22:22:45 -0400
On Tue, 11 Sep 2007 16:25:30 -0700, David R Tribble
<david@xxxxxxxxxxx> wrote:
tommy1729 wrote:
tommy's conjecture:
(1 = prime)
every positive integer is the sum of at most 8 times primes^2
How is this more impressive than Goldbach's Conjecture?
If Golbach's Conjecture is true, then every positive integer
is either the sum of two primes (p1+p2, even) or the sum
of two primes plus one (p1+p2+1, odd).
Tommy's conjecture relates to sums of _squares_ of primes.
He conjectures that every positive integer is a sum of at most 8
squares of primes (where primes are extended to include 1).
Why 8 and not less? I don't know. I'm sure he has his reasons.
3 is too obviously too low, since a number of the form 8k+7 is not a
sum of 3 or fewer squares of positive integers, much less primes.
Is 4 too low? I'm not sure. But even if there are counterexamples, the
more significant question is whether there are there infinitely many
such counterexamples.
I think the related density question is also interesting, and possibly
within reach of currently known methods:
How many squares of primes (or 1) need to be summed to reach density
1? I suspect at most 5, but I could be wrong. However, I'm pretty sure
it's less than 8.
quasi
.
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