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Prime Representation


Let a!=b, A, and B denote positive integers satisfying

 (a,b)=1    (A,B)=1,

(i.e., both pairs are relatively prime), and suppose every prime p=B (mod A) 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=-B (mod A) 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 formrepresentation
4n+1x^2+y^2
8n+1,8n+3x^2+2y^2
8n+/-1x^2-2y^2
6n+1x^2+3y^2
12n+1x^2-3y^2
20n+1,20n+9x^2+5y^2
10n+1,10n+9x^2-5y^2
14n+1,14n+9,14n+25x^2+7y^2
28n+1,28n+9,28n+25x^2-7y^2
30n+1,30n+49x^2+15y^2
60n+1,60n+49x^2-15y^2
30n-7,30n+175x^2+3y^2
60n-7,60n+175x^2-3y^2
24n+1,24n+7x^2+6y^2
24n+1,24n+19x^2-6y^2
24n+5,24n+112x^2+3y^2
24n+5,24n-12x^2-3y^2

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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.

Referenced on Wolfram|Alpha

Prime Representation

Cite this as:

Weisstein, Eric W. "Prime Representation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PrimeRepresentation.html

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