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Prime Diophantine Equations

k+2 is prime iff the 14 Diophantine equations in 26 variables

wz+h+j-q=0
(1)
(gk+2g+k+1)(h+j)+h-z=0
(2)
16(k+1)^3(k+2)(n+1)^2+1-f^2=0
(3)
2n+p+q+z-e=0
(4)
e^3(e+2)(a+1)^2+1-o^2=0
(5)
(a^2-1)y^2+1-x^2=0
(6)
16r^2y^4(a^2-1)+1-u^2=0
(7)
n+l+v-y=0
(8)
(a^2-1)l^2+1-m^2=0
(9)
ai+k+1-l-i=0
(10)
{[a+u^2(u^2-a)]^2-1}(n+4dy)^2+1-(x+cu)^2=0
(11)
p+l(a-n-1)+b(2an+2a-n^2-2n-2)-m=0
(12)
q+y(a-p-1)+s(2ap+2a-p^2-2p-2)-x=0
(13)
z+pl(a-p)+t(2ap-p^2-1)-pm=0
(14)

have a solution in positive integers (Jones et al. 1976; Riesel 1994, p. 40).

See also

Prime-Generating Polynomial

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References

Jones, J. P.; Sato, D.; Wada, H.; and Wiens, D. "Diophantine Representation of the Set of Prime Numbers." Amer. Math. Monthly 83, 449-464, 1976.Riesel, H. Prime Numbers and Computer Methods for Factorization, 2nd ed. Boston, MA: Birkhäuser, 1994.

Referenced on Wolfram|Alpha

Prime Diophantine Equations

Cite this as:

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

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