I wrote:

>"Mark Underwood" <mark.underwood@...> wrote:

>

>>>>It looks like all primes can also be written as a^2 + b^2 + 2c^2

>>>

>>> Legendre proved that all positive odd integers can be represented

>>>in that form - see e.g. page 3 of:

>>> www.math.wisc.edu/~ono/reprints/025.pdf

>>>

>>

>>Yes, a guru has [...] also

>>conjectured that 1a^2 + 2b^2 + 3c^2 itself can represent all odd

>>positive integers.

>>

>

>If we give up the restriction to *odd* integers, it is interesting to find that some (even) numbers can't be represented.

>

>This is fully in accord with the remark on p.6 of the following excellent paper:-

>http://www.pubmedcentral.gov/articlerender.fcgi?artid=1076292

>namely: "Universal forms - A positive form representing all positive integers .. is said to be universal. Methods of the previous section show that no positive ternary quadratic form is universal."

>

>So there were *bound* to be integers not representable.

>

>Up to 200, the list is just the following:-

>10=2*5 = 2 mod 8

>26=2*13 = 2 mod 8

>40=2^3*5 = 0 mod 8

>42=2*3*7 = 2 mod 8

>58=2*29 = 2 mod 8

>74=2*37 = 2 mod 8

>90=2*3^2*5 = 0 mod 8

>104=2^3*13 = 0 mod 8

>106=2*53 = 2 mod 8

>122=2*61 = 2 mod 8

>138=2*3*23 = 2 mod 8

>154=2*7*11 = 2 mod 8

>160=2^5*5 = 0 mod 8

>168=2^3*3*7 = 0 mod 8

>170=2*7*17 = 2 mod 8

>186=2*3*31 = 2 mod 8

>

>Note the remarkable restriction to 0 or 2 mod 8 !

>Other than that, there doesn't appear to be any obvious pattern.

That was for the form a^2 + 2*b^2 + 3*c^2.

It's interesting to explore also the exceptions to the form

a^2 + b^2 + 2*c^2 (*)

In this case there *does* seem to be a regular pattern, and moreover one which is understandable.

Consider the form (*) modulo 16.

Any odd integer has its square = 1 or 9 mod 16,

any even integer has square = 0 or 4 mod 16.

So (a^2 + b^2) = {0, 1, 2, 4, 5, 8, 9, 10, 13} mod 16,

and (2*c^2) = {0, 2, 8} mod 16.

So (*) <> 14 mod 16.

Clearly, multiplying a, b and c by 2,

we have also that

(*) <> 56 mod 64,

(*) <> 224 mod 256,

and so on.

I have checked up to 760, and there are no other exceptions.

Would anyone like to check up to some high bound that this is the *complete* set of unrepresetable integers?

Can anyone *prove* same? [This is likely to be *very* hard!]

Does anyone know a web page that gives Legendre's proof that all *odd* integers are representable?

(What a really clever guy he was!)

-Mike Oakes