In number theory, Heegner numbers are square-free positive integers d {\displaystyle d} such that the imaginary quadratic field Q ( − d ) {\displaystyle \mathbb {Q} ({\sqrt {-d}})} has class number 1. Equivalently, the ring of algebraic integers of Q ( − d ) {\displaystyle \mathbb {Q} ({\sqrt {-d}})} has unique factorization. The determination of such numbers is a special case of the class number problem, and they underlie several striking results in number theory. According to the Stark–Heegner theorem there are precisely nine Heegner numbers:
This result was conjectured by Gauss and proved up to minor flaws by Kurt Heegner in 1952. Alan Baker and Harold Stark independently proved the result in 1966, and Stark further indicated that the gap in Heegner's proof was minor.
Euler's prime-generating polynomial Euler's prime-generating polynomial
n 2 + n + 41 {\displaystyle n^{2}+n+41} , which gives distinct primes for n = 0 , . . . , 39 {\displaystyle n=0,...,39} , is related to the Heegner number 163 = 4 ⋅ 41 − 1 {\displaystyle 163=4\cdot 41-1} by the discriminant of the polynomial. Rabinowitsch proved that the polynomial
n 2 + n + p {\displaystyle n^{2}+n+p}
gives primes for n = 0 , … , p − 2 {\displaystyle n=0,\dots ,p-2} if and only if its discriminant 1 − 4 p {\displaystyle 1-4p} is the negative of a Heegner number. 1, 2, and 3 are not of the required form, so the Heegner numbers that work are 7, 11, 19, 43, 67, 163, yielding prime generating functions of Euler's form for 2, 3, 5, 11, 17, 41; these latter numbers are called lucky numbers of Euler by F. Le Lionnais.
Almost integers and Ramanujan's constant Ramanujan's constant is the transcendental number
e π 163 {\displaystyle e^{\pi {\sqrt {163}}}} , which is an almost integer:
e π 163 = 262 537 412 640 768 743.999 999 999 999 25 … ≈ 640 320 3 + 744. {\displaystyle e^{\pi {\sqrt {163}}}=262\,537\,412\,640\,768\,743.999\,999\,999\,999\,25\ldots \approx 640\,320^{3}+744.}
This number was discovered in 1859 by the mathematician Charles Hermite. In a 1975 April Fools' Day article in Scientific American magazine, "Mathematical Games" columnist Martin Gardner made the hoax claim that the number was in fact an integer, and that the Indian mathematical genius Srinivasa Ramanujan had predicted it—hence its name.
Details This coincidence is explained by complex multiplication and the q-expansion of the j-invariant. In what follows, j ( z ) {\displaystyle j(z)} denotes the j-invariant of the complex number z {\displaystyle z} . Briefly, j ( 1 + − d 2 ) {\displaystyle \textstyle j\left({\frac {1+{\sqrt {-d}}}{2}}\right)} is an integer for d a Heegner number, and
e π d ≈ − j ( 1 + − d 2 ) + 744 {\displaystyle e^{\pi {\sqrt {d}}}\approx -j\left({\frac {1+{\sqrt {-d}}}{2}}\right)+744}
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