Are you including 2 as a prime? --Rich ________________________________________ From: math-fun [math-fun-bounces@mailman.xmission.com] on behalf of David Wilson [davidwwilson@comcast.net] Sent: Sunday, September 11, 2016 8:09 PM To: 'math-fun' Subject: [EXTERNAL] Re: [math-fun] Eight prime square theorem? Yes. If 1 is not admitted as a square prime, a few values <= 23 cannot be represented. The numbers requiring 8 terms starts off 32, 795, 803, 811, ... and seems to slowly grow denser. On the range 0 <= n <= 10^7, 334119 values require 8 terms, the last being 9999937. No values require 9 or more terms. On that basis, it seems reasonable to conjecture that 8 terms is necessary and sufficient.
-----Original Message----- From: math-fun [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Dan Asimov Sent: Thursday, September 08, 2016 1:58 AM To: math-fun Subject: Re: [math-fun] Eight prime square theorem?
Very interesting! (Which of course also brings up the question of sums of kth prime powers for any given k.)
Dumb question: What is the exact meaning of "8 squares of primes are necessary for certain arbitrarily large numbers" ?
Does this mean the set of numbers requiring eight prime squares is unbounded ? Or something else?
—Dan
On Sep 7, 2016, at 5:47 PM, David Wilson <davidwwilson@comcast.net> wrote:
If 1 is not counted as the square of a prime, then nonnegative numbers
1, 2, 3, 5, 6, 7, 10, 11, 14, 15, 19, 23
are not a sum of squares of primes (specifically, not a sum of terms equal to 4 or 9).
It looks as if at most 8 squares of primes are sufficient to add to any other nonnegative number, and are necessary for certain arbitrarily large numbers.
-----Original Message----- From: math-fun [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of David Wilson Sent: Wednesday, September 07, 2016 8:30 PM To: 'math-fun' Subject: [math-fun] Eight prime square theorem?
Lagrange's four square theorem states that every nonnegative integer is a sum of four squares. Empirically, it looks as if every nonnegative integer is a sum of 8 squares of primes (counting 1 as prime). It looks as if arbitrarily large numbers require 8 terms.
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