Re: representation

In article <4ote63hjc09h78iorf40qjemekfqjbnsk5@xxxxxxx>,
quasi <quasi@xxxxxxxx> wrote:

On Wed, 06 Jun 2007 17:29:09 -0700, slapjack777@xxxxxxxxxxx wrote:

On Jun 6, 3:37 pm, tommy1729 <tommy1...@xxxxxxxxx> wrote:
every positive integer is the sum of at most 8
squared primes (including 1)

tommy1729

cmon people show your math skills :-)

not even a crankpot gonna try ??

nobody this year ?

http://mathworld.wolfram.com/LagrangesFour-SquareTheorem.html

To say that every positive integer is a sum of 4 squares is not the
same as saying every positive integer is the sum of at most 8 squares
of primes (including 1).

Tommy1729's sum of 8 squares of primes conjecture is a reasonable
claim, but needs some numerical testing.

Of course, a single counterexample would kill it instantly.

Another way to disprove the conjecture without producing an explicit
counterexample would be to show that the density of the set of numbers
which can be represented as sum of at most 8 squares of primes
(including 1) is less than 1.

On the other hand, if the conjecture is true, proving it might be near
impossible.

However, even if 8 doesn't work, it's intuitive that it will work for
_some_ positive integer k in place of 8.

So here's my more modest version of Tommy1729's conjecture ...

Conjecture (1):

For every positive integer n, there is a positive integer k, depending
only on n, such that every positive integer is a sum of at most k n'th
powers of primes (where we include 1 as a prime).

Here's an even weaker conjecture ...

Conjecture (2):

For every positive integer n, there is a positive integer k, depending
only on n, such that the density of the set of positive integers which
can be represented as a sum of at most k n'th powers of primes (where
we include 1 as a prime) is 1.

Remarks:

Clearly, if conjecture (2) fails for a given n, then it fails for all
higher values of n. Moreover, if conjecture (2) fails for a given n ,
then conjecture (1) also fails for that n.

For n=1, both conjectures are almost certainly true, and are perhaps
instantly implied by known results.

Moreover, for n=1, since k is free to be any fixed positive integer,
there might be really elementary proofs, at least for conjecture (2).

Of course, for n=1, the truth of Goldbach's conjecture would imply
that k=3 suffices (and hence is best possible) for conjecture (1).

Note that for n=2, Tommy1729 claims that k=8 suffices for conjecture
(1), whereas I only claim that some k suffices.

quasi

This may be of interest.

MR1775322 (2001i:11119)
Liu, Jianya(PRC-SHAN); Liu, Ming-Chit(PRC-HK)
The exceptional set in the four prime squares problem. (English summary)
Illinois J. Math. 44 (2000), no. 2, 272--293.
11P32 (11P55)

It is conjectured that every sufficiently large integer $n\equiv
4\pmod{24}$ is a sum of the squares of four primes. The present paper
shows that amongst the integers $n\leq N$ there are at most
$O(N^{13/15+\varepsilon})$ exceptions to this conjecture, for any
$\varepsilon>0$. Previously W. Schwarz \ref[J. Reine Angew. Math. 206
(1961), 78--112; MR0126431 (23 \#A3727)] had shown that there can be at
most $O(N(\log N)^{-A})$ exceptions, for any $A>0$. The proof hinges on
a sharp "major arcs" estimate. Here the major arcs are centred on
fractions $a/q$ with $q\leq N^{2/15-\varepsilon}$. The ability to
consider such a substantial range for $q$ is crucial. There are no
problems concerned with exceptional real zeros of $L$-functions, nor
with numerical constants in the formulation of zero-free regions for
$L$-functions. Thus the analysis is considerably simpler, and can be
pushed rather further than that arising for the exceptional set for the
Goldbach problem, as considered by H. L. Montgomery and R. C. Vaughan
\ref[Acta Arith. 27 (1975), 353--370; MR0374063 (51 \#10263)].

Reviewed by D. R. Heath-Brown

--
Gerry Myerson (gerry@xxxxxxxxxxxxxxx) (i -> u for email)
.

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