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Sunday, October 2, 2016

Entry 34

Here are some optimum prime-generating polynomials P(n)=n2+n+41P(n)=2n2+29P(n)=2n2+2n+19P(n)=3n2+3n+23P(n)=5n2+5n+13P(n)=6n2+6n+31P(n)=7n2+7n+17The first one is the most famous, being Euler's. Expressed as the quadratic P(n)=an2+bn+c, its discriminant is d=b24ac. Using the values a,d of the above, one gets
eπ/1163=6403203+743.9999999999992eπ/2232=3964104.0000001eπ/2148=(842)4+103.99997eπ/3267=3003+41.99997eπ/55×47=(1847)2+15.991eπ/66×118=10602+9.99992eπ/77×61=(397)2+9.995These approximations (actually, the exact values of certain eta quotients) can be used as denominators in pi formulas known as Ramanujan-Sato series to be discussed in Entry 35.
 

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