Neil Sloane asked me about commands in computer languages to find the (positive) primes represented by indefinite binary quadratic forms. So I wrote something in C++ that works. This is for the OEIS, these primes go into sequences... Note that, within a few hours, another guy had run the tables much higher with a one-line Maple command. Some days it does not pay to get up.

I thought of one I really do not understand. Discriminant $205$ has four classes of forms, $$ \langle 1, 13, -9 \rangle, \; \langle -1, 13, 9 \rangle, \; \langle 3, 13, -3 \rangle, \; \langle -3, 13, 3 \rangle. $$

The third and fourth are opposites so in the same genus, although distinct. The first two are in the principal genus, but they are not opposites, one is $-1$ times the other; in particular, they get diffeent positive primes, although both do residues $\pmod 5$ and $\pmod {41}.$ For $\langle 1, 13, -9 \rangle$ we get $$ 1,5,59,131,139,241,269,271,359,409, \ldots, $$ while for $\langle -1, 13, 9 \rangle$ we get $$ 31,41,61,251,349,379,389,401,419,431, \ldots. $$

For positive forms, low class number, there are polynomials, such as in Cox's book, such that primes represented by the principal form are those for which the polynomial factors a certain way. For a prime $p \equiv 1 \pmod 3,$ Gauss showed that $2$ is a cubic residue if an only if $p = u^2 + 27 v^2.$ Jacobi showed that $3$ is a cubic residue if an only if $p = u^2 + uv + 61 v^2.$ All I found in Henri Cohen's tables was the fact that $\mathbb Q(\sqrt {205})$ has class number $2$ and $L_K = \mathbb Q(\sqrt 5),$ appendix 12C on pages 533 and 534.

Let's see, $34$ is the smallest number where it is a surprise that there is no solution to $x^2 - 34 y^2 = -1.$ The smallest such odd number is $205,$ as there is no solution to $x^2 - 205 y^2 = -1.$ For prime $p \equiv 1 \pmod 4,$ there is always a solution to $x^2 - p y^2 = -1.$ Proof in Mordell's book. Anyway, this is why $\langle 1, 13, -9 \rangle, \; \langle -1, 13, 9 \rangle$ are distinct classes.

So, that is the question, can I distinguish the represented (positive) primes by factoring some polynomial mod these primes?

Neil Sloane asked me about commands in computer languages to find the (positive) primes represented by indefinite binary quadratic forms. So I wrote something in C++ that works. This is for the OEIS, these primes go into sequences... Note that, within a few hours, another guy had run the tables much higher with a one-line Maple command. Some days it does not pay to get up.

I thought of one I really do not understand. Discriminant $205$ has four classes of forms, $$ \langle 1, 13, -9 \rangle, \; \langle -1, 13, 9 \rangle, \; \langle 3, 13, -3 \rangle, \; \langle -3, 13, 3 \rangle. $$

The third and fourth are opposites so in the same genus, although distinct. The first two are in the principal genus, but they are not opposites, one is $-1$ times the other; in particular, they get diffeent positive primes, although both do residues $\pmod 5$ and $\pmod {41}.$ For $\langle 1, 13, -9 \rangle$ we get $$ 1,5,59,131,139,241,269,271,359,409, \ldots, $$ while for $\langle -1, 13, 9 \rangle$ we get $$ 31,41,61,251,349,379,389,401,419,431, \ldots. $$

For positive forms, low class number, there are polynomials, such as in Cox's book, such that primes represented by the principal form are those for which the polynomial factors a certain way. For a prime $p \equiv 1 \pmod 3,$ Gauss showed that $2$ is a cubic residue if an only if $p = u^2 + 27 v^2.$ Jacobi showed that $3$ is a cubic residue if an only if $p = u^2 + uv + 61 v^2.$ All I found in Henri Cohen's tables was the fact that $\mathbb Q(\sqrt {205})$ has class number $2$ and $L_K = \mathbb Q(\sqrt 5),$ appendix 12C on pages 533 and 534. See related information at IT'S A LINK.

Let's see, $34$ is the smallest number where it is a surprise that there is no solution to $x^2 - 34 y^2 = -1.$ The smallest such odd number is $205,$ as there is no solution to $x^2 - 205 y^2 = -1.$ For prime $p \equiv 1 \pmod 4,$ there is always a solution to $x^2 - p y^2 = -1.$ Proof in Mordell's book. Anyway, this is why $\langle 1, 13, -9 \rangle, \; \langle -1, 13, 9 \rangle$ are distinct classes.

So, that is the question, can I distinguish the represented (positive) primes by factoring some polynomial mod these primes?

Neil Sloane asked me about commands in computer languages to find the (positive) primes represented by indefinite binary quadratic forms. So I wrote something in C++ that works. This is for the OEIS, these primes go into sequences... Note that, within a few hours, another guy had run the tables much higher with a one-line Maple command. Some days it does not pay to get up.

I thought of one I really do not understand. Discriminant $205$ has four classes of forms, $$ \langle 1, 13, -9 \rangle, \; \langle -1, 13, 9 \rangle, \; \langle 3, 13, -3 \rangle, \; \langle -3, 13, 3 \rangle. $$

The third and fourth are opposites so in the same genus, although distinct. The first two are in the principal genus, but they are not opposites, one is $-1$ times the other; in particular, they get diffeent positive primes, although both do residues $\pmod 5$ and $\pmod {41}.$ For $\langle 1, 13, -9 \rangle$ we get $$ 1,5,59,131,139,241,269,271,359,409, \ldots, $$ while for $\langle -1, 13, 9 \rangle$ we get $$ 31,41,61,251,349,379,389,401,419,431, \ldots. $$

For positive forms, low class number, there are polynomials, such as in Cox's book, such that primes represented by the principal form are those for which the polynomial factors a certain way. For a prime $p \equiv 1 \pmod 3,$ Gauss showed that $2$ is a cubic residue if an only if $p = u^2 + 27 v^2.$ Jacobi showed that $3$ is a cubic residue if an only if $p = u^2 + uv + 61 v^2.$ All I found in Henri Cohen's tables was the fact that $\mathbb Q(\sqrt {205})$ has class number $2$ and $L_K = \mathbb Q(\sqrt 5),$ appendix 12C on pages 533 and 534.

So, that is the question, can I distinguish the represented (positive) primes by factoring some polynomial mod these primes?

Neil Sloane asked me about commands in computer languages to find the (positive) primes represented by indefinite binary quadratic forms. So I wrote something in C++ that works. This is for the OEIS, these primes go into sequences... Note that, within a few hours, another guy had run the tables much higher with a one-line Maple command. Some days it does not pay to get up.

I thought of one I really do not understand. Discriminant $205$ has four classes of forms, $$ \langle 1, 13, -9 \rangle, \; \langle -1, 13, 9 \rangle, \; \langle 3, 13, -3 \rangle, \; \langle -3, 13, 3 \rangle. $$

The third and fourth are opposites so in the same genus, although distinct. The first two are in the principal genus, but they are not opposites, one is $-1$ times the other; in particular, they get diffeent positive primes, although both do residues $\pmod 5$ and $\pmod {41}.$ For $\langle 1, 13, -9 \rangle$ we get $$ 1,5,59,131,139,241,269,271,359,409, \ldots, $$ while for $\langle -1, 13, 9 \rangle$ we get $$ 31,41,61,251,349,379,389,401,419,431, \ldots. $$

For positive forms, low class number, there are polynomials, such as in Cox's book, such that primes represented by the principal form are those for which the polynomial factors a certain way. For a prime $p \equiv 1 \pmod 3,$ Gauss showed that $2$ is a cubic residue if an only if $p = u^2 + 27 v^2.$ Jacobi showed that $3$ is a cubic residue if an only if $p = u^2 + uv + 61 v^2.$ All I found in Henri Cohen's tables was the fact that $\mathbb Q(\sqrt {205})$ has class number $2$ and $L_K = \mathbb Q(\sqrt 5),$ appendix 12C on pages 533 and 534.

Let's see, $34$ is the smallest number where it is a surprise that there is no solution to $x^2 - 34 y^2 = -1.$ The smallest such odd number is $205,$ as there is no solution to $x^2 - 205 y^2 = -1.$ For prime $p \equiv 1 \pmod 4,$ there is always a solution to $x^2 - p y^2 = -1.$ Proof in Mordell's book. Anyway, this is why $\langle 1, 13, -9 \rangle, \; \langle -1, 13, 9 \rangle$ are distinct classes.

So, that is the question, can I distinguish the represented (positive) primes by factoring some polynomial mod these primes?