Re: TOMMYS CONJECTURE sorry quasi way above 5.

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.

of course , we already need 4 squares to represent all integers

But even if there are
counterexamples

there are , they are in my sequence given below


, the
more significant question is whether there are there
infinitely many
such counterexamples.

yep infinitely many.



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.

nope 8.

quasi

see my sequence:

http://www.research.att.com/~njas/sequences/A096436

notice 795 requires 8.

and already 73 requires more than 5.

robert g wilson bets on =< 9

regards
tommy1729
.

Relevant Pages

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