Sums of a square and twice a square

[Allan Wechsler]

Numbers of the form x^2 + 2(y^2) are listed at oeis.org/A002479.

It seemed natural to me to look for numbers that can be expressed in this form two different ways. I found 9, 18, 27, 33, 36, 51 ... and already could see that this sequence was not in OEIS. I feel like I must have made a stupid arithmetic error, because the idea is so simple, so if somebody can spot my mistake I'd appreciate it.

I noticed that all such numbers I found were of the form 3k, and I conjecture that all numbers of the form x^2 + 2(y^2) in more than one way are divisible by 3.

I looked at one-third of my sequence, 3, 6, 9, 11, 12, 17, ... and found it at oeis.org/A154777. These are, weirdly, the numbers of the form x^2 + 2(y^2) where x and y are positive. (I allowed zero in my original list.) A second conjecture: if k is in A154777, then 3k can be expressed as x^2 + 2(y^2) in more than one way.

If k is in A002479, is 3k also? It looks like this is true; let's call it a third conjecture.

I really ought to learn more about quadratic forms; I am woefully ignorant. I would appreciate any proofs or counterexamples.

The table of x^2 + 2(y^2) read by diagonals is not in OEIS either. The x^2 + y^2 version is at oeis.org/A048147.

-- Allan

[jpallouche.math]

Hi

From A002479 "Euler (E256) shows that these numbers are closed under multiplication" (a proof can be obtained by mimicking the corresponding proof for x^2 + y^2. Since 3 = 1^2 + 2(1^2) your "third conjecture" is true. best jean-paul

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[Gareth McCaughan]

On 14/05/2026 18:43, Allan Wechsler wrote:

Numbers of the form x^2 + 2(y^2) are listed at oeis.org/A002479.

It seemed natural to me to look for numbers that can be expressed in this form two different ways. I found 9, 18, 27, 33, 36, 51 ... and already could see that this sequence was not in OEIS. I feel like I must have made a stupid arithmetic error, because the idea is so simple, so if somebody can spot my mistake I'd appreciate it.

I noticed that all such numbers I found were of the form 3k, and I conjecture that all numbers of the form x^2 + 2(y^2) in more than one way are divisible by 3.

This is not true; 323 = 9^2 + 2.11^2 = 15^2 + 2.7^2 is not divisible by 3.

I looked at one-third of my sequence, 3, 6, 9, 11, 12, 17, ... and found it at oeis.org/A154777. These are, weirdly, the numbers of the form x^2 + 2(y^2) where x and y are positive. (I allowed zero in my original list.) A second conjecture: if k is in A154777, then 3k can be expressed as x^2 + 2(y^2) in more than one way.

This is true, as is something stronger: the product of any two numbers in A154777 (one of which is 3) is expressible as x^2+2y^2 in more than one way.

Proof: write s=sqrt(-2) and consider numbers of the form x+ys where x,y are integers; write conj(x+ys) = x-ys and norm(x+ys) = x^2+2y^2 = (x+ys).conj(x+ys); note that conj(ab)=conj(a)conj(b) and hence norm(ab)=norm(a)norm(b); now A154777 consists of the norms of things x+ys where x,y are both nonzero; if a,b are both of this form then norm(ab) = norm(a.conj(b)) displays two equal things of form x^2+2y^2 with different x,y.

(The example I gave earlier was obtained by picking two smallish non-multiple-of-3 entries from the list, namely 17 and 19.)

If k is in A002479, is 3k also? It looks like this is true; let's call it a third conjecture.

As Jean-Paul Allouche has observed, this is true; more generally, the product of two things in A002479 is always in A002479. The ones that aren't also in A154777 don't give rise to multiple representations as x^2+2y^2 because for such numbers the conjugate of x+ys is not "interestingly" different from x+ys itself.

I think the following is true: the numbers that are x^2+2y^2 in more than one way are _precisely_ the products of numbers in A154777. The key observation here is that any (rational) prime number either is still prime in Z[sqrt(-2)] or is the product of a pair of conjugates in Z[sqrt(-2)] in exactly one way. But I am not an expert on this and may be missing some edge cases or something.

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[Sean A. Irvine]

Hi Allan,

Also relevant: A002325 which gives the number of solutions.

Sean.

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