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Prime Representation

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

\begin{displaymath}
(a,b)=1\qquad (A,B)=1,
\end{displaymath}

(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.




© 1996-9 Eric W. Weisstein
1999-05-26