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Joe Roberts writes, in Lure of the Integers, that Matijasevič showed that "every integer has a representation in the form $a^2+b^2+c^2+c+1$". The citation he gives is

Ju. V. Matijasevič, A Diophantine representation of the set of prime numbers (Russian), Dokl. Akad. Nauk SSSR 196 (1971) 770-773.

(Surely the intent was that all positive integers are of the indicated form, where $a,b,c$ can take on nonnegative integer values.) I was able to find an English summary of the paper

useless summary of the paper

but it explained only the main result, not this (apparently) ancillary result. What is known about this result and other polynomials representing all positive integers? (Aside from all the standard four-square stuff, that is.) Is there an English-language proof, preferably simplified from the original?

Charles
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2 Answers2

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If you multiply by 4 you get $$ 4 n = 4 a^2 + 4 b^2 + (2c+1)^2 + 3, $$ or $$ 4 n - 3 = 4 a^2 + 4 b^2 + (2c+1)^2 . $$ As it happens, the ternary form $$ u^2 + 4 v^2 + 4 w^2 $$ is regular in the sense of Dickson and does represent all numbers $ \equiv 1 \pmod 4.$ So this result long predates your author, it was known by 1939 and was probably known to Gauss. See Dickson_Diagonal at ME

There is a fair industry now in not-quite-homogeneous polynomials of degree no higher than two representing all or almost all positive integers. A new Ph. D., very active in this, is Anna Haensch. Her adviser is Wai Kiu Chan of Wesleyan. Chan has also published on this with Byeong-Kweon Oh of Seoul National. Let's see, as far as actual quadratic forms, Hanke and Bhargava are finally going to publish the 290 theorem article, but the facts will be what people already know: a positive quadratic form represents all positive numbers if and only if it represents (integrally) all the numbers from 1 to 290 inclusive.

Finally, there are some surprises if higher degree terms are allowed. By density arguments, you would expect $x^2 + y^2 + z^9$ to integrally represent all, or all sufficiently large, positive integers, but in fact $$ x^2 + y^2 + z^9 \neq 216 p^3 $$ for any positive prime $p \equiv 1 \pmod 4.$

Will Jagy
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15 and 290 theorems states that if an integral quadratic form with integer matrix represents all positive integers up to 15, then it represents all positive integers.

lsr314
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  • Thanks for the answer (+1). I do know about the results on homogeneous polynomials, including the 15 and 290 theorems. – Charles Jul 30 '13 at 23:50