eπ/1√163=6403203+743.9999999999992…eπ/2√232=3964−104.0000001…eπ/2√148=(84√2)4+103.99997…eπ/3√267=3003+41.99997…eπ/5√5×47=(18√47)2+15.991…eπ/6√6×118=10602+9.99992…eπ/7√7×61=(39√7)2+9.995…These 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.
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=b2−4ac.
Using the values a,d of the above, one gets
eπ/1√163=6403203+743.9999999999992…eπ/2√232=3964−104.0000001…eπ/2√148=(84√2)4+103.99997…eπ/3√267=3003+41.99997…eπ/5√5×47=(18√47)2+15.991…eπ/6√6×118=10602+9.99992…eπ/7√7×61=(39√7)2+9.995…These 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.
eπ/1√163=6403203+743.9999999999992…eπ/2√232=3964−104.0000001…eπ/2√148=(84√2)4+103.99997…eπ/3√267=3003+41.99997…eπ/5√5×47=(18√47)2+15.991…eπ/6√6×118=10602+9.99992…eπ/7√7×61=(39√7)2+9.995…These 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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