Prime Representation

Let $a\not=b$, $A$, and $B$ denote Positive Integers satisfying

$$(a,b)=1\qquad (A,B)=1,$$

(i.e., both pairs are Relatively Prime), and suppose every Prime $p\equiv B\ \left({{\rm mod\ } {A}}\right)$ with $(p,2ab)=1$ is expressible if the form $ax^2-by^2$ for some Integers $x$ and $y$. Then every Prime $q$ such that $q\equiv -B\ \left({{\rm mod\ } {A}}\right)$ and $(q,2ab)=1$ is expressible in the form $bX^2-aY^2$ for some Integers $X$ and $Y$ (Halter-Koch 1993, Williams 1991).

Prime Form	Representation
$4n+1$	$x^2+y^2$
$8n+1, 8n+3$	$x^2+2y^2$
$8n\pm 1$	$x^2-2y^2$
$6n+1$	$x^2+3y^2$
$12n+1$	$x^2-3y^2$
$20n+1, 20n+9$	$x^2+5y^2$
$10n+1, 10n+9$	$x^2-5y^2$
$14n+1, 14n+9, 14n+25$	$x^2+7y^2$
$28n+1, 28n+9, 28n+25$	$x^2-7y^2$
$30n+1, 30n+49$	$x^2+15y^2$
$60n+1, 60n+49$	$x^2-15y^2$
$30n-7, 30n+17$	$5x^2+3y^2$
$60n-7, 60n+17$	$5x^2-3y^2$
$24n+1, 24n+7$	$x^2+6y^2$
$24n+1, 24n+19$	$x^2-6y^2$
$24n+5, 24n+11$	$2x^2+3y^2$
$24n+5, 24n-1$	$2x^2-3y^2$

References

Berndt, B. C. Ramanujan's Notebooks, Part IV. New York: Springer-Verlag, pp. 70-73, 1994.

Halter-Koch, F. ``A Theorem of Ramanujan Concerning Binary Quadratic Forms.'' J. Number. Theory 44, 209-213, 1993.

Williams, K. S. ``On an Assertion of Ramanujan Concerning Binary Quadratic Forms.'' J. Number Th. 38, 118-133, 1991.