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Definition :

Idoneal numbers are the positive integers $D$ such that any integer expressible in only one way as $x^2 ± Dy^2$ (where $x^2$ is relatively prime to $Dy^2$) is a prime, prime power, or twice one of these.

Number $32$ satisfies this definition because :

$p^2 = x^2+32 \cdot y^2$ if and only if : $p \equiv 1 \pmod {32}$ or $p \equiv 17 \pmod {32}$ , and every prime number

p is expressible in exactly one way as : $\sqrt{ x^2+32 \cdot y^2}$ ,where $\gcd(x^2,32 \cdot y^2)=1$ .

However , there is an additional condition on Idoneal numbers :

"A positive integer $D$ is idoneal iff it cannot be written as $ab+bc+ca$ for integer $a, b$, and $c$

with $0<a<b<c$."

Since , $32 = 1 \cdot 2 + 2 \cdot 10 +1 \cdot 10$ number $32$ doesn't satisfy this condition and therefore it isn't Idoneal number .

My question : Why definition of Indoneal numbers is inconsistent with the $abc$ requirement in case of number $32$ ?

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3 Answers

up vote 4 down vote accepted

The discussion so far is a little backwards -- the question of non-idoneality is not about representability of primes. To prove non-idoneality of $D$, one has to demonstrate an $n$ which is uniquely expressible in the form $x^2+Dy^2$ which is not a prime power or twice a prime power.

In your case, $D=32$ is not idoneal because the $n=33$ is not a prime, prime power, or twice a prime power but is expressible uniquely in the form $x^2+32y^2$. As you observe, this is consistent with this alternative characterization of idoneal numbers in terms of expressibility by the form $ab+ac+bc$.

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Condition for $n$ to be idoneal from OEIS : (2) Every genus of quadratic forms of discriminant -4n consists of a single class. [Gauss]

The principal genus consists of two classes, $$ f(x,y) = x^2 + 32 y^2 $$ and $$ g(x,y) = 4 x^2 + 4 x y + 9 y^2. $$ The other two forms of discriminant $-128$ are $$ 3 x^2 \pm 2 x y + 11 y^2 $$

All primes $p \equiv 1 \pmod 8$ are represented by either $f$ or $g.$ The primes up to 1200 that are represented by $x^2 + 32 y^2$ are

41 113 137 257 313 337 353 409 457 521 569 577 593 761 809 857 881 953 1129 1153

The primes up to 1200 that are represented by $4x^2 +4 x y + 9 y^2$ are

17 73 89 97 193 233 241 281 401 433 449 601 617 641 673 769 929 937 977 1009 1033 1049 1097 1193

Once $p \equiv 1 \pmod 8,$ there is an integral representation $p = x^2 + 32 y^2$ if and only if there is an integer solution $z$ to $$ z^4 - 2 z^2 + 2 \equiv 0 \pmod p $$

If $p \equiv 1 \pmod 8,$ then $z^4 - 2 z^2 + 2$ has four distinct linear factors $\pmod p$ when $p = x^2 + 32 y^2,$ but two quadratic factors and no roots $\pmod p$ when $p = 4x^2 +4 x y + 9 y^2.$ When $p \equiv 3 \pmod 8,$ then $z^4 - 2 z^2 + 2$ is irreducible $\pmod p.$

The other odd primes are not represented by anything of this discriminant. If $p \equiv 7 \pmod 8,$ then $z^4 - 2 z^2 + 2$ has two quadratic factors and no roots $\pmod p.$ If $p \equiv 5 \pmod 8,$ then $z^4 - 2 z^2 + 2$ has two linear and one quadratic factor, for two roots $\pmod p.$

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Note that you have NOT proved that EVERY prime number $p$ is uniquely expressible as $\sqrt{x^2+32y^2}$, where $(x^2,32y^2)=1$. You've simply stated what you wish to prove $:)$

While the relation

$p^2 = x^2+32 \cdot y^2$ if and only if : $p \equiv 1 \pmod {32}$ or $p \equiv 17 \pmod {32}$

is true, this does NOT prove the uniqueness of $x,y$ satisfying above condition for a given $p$. So you cannot claim that there is some inconsistency in the first definition of Idoneal numbers.

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Note that you have not disproved uniqueness... –  pedja Jan 10 '12 at 10:27
    
true, I have not disproved uniqueness. I was pointing out that your argument is not conclusive in showing inconsistency in definition of Idoneal numbers. –  Nikhil Bellarykar Jan 10 '12 at 10:31
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