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}
via the q-expansion. If τ {\displaystyle \tau } is a quadratic irrational, then its j-invariant j ( τ ) {\displaystyle j(\tau )} is an algebraic integer of degree | C l ( Q ( τ ) ) | {\displaystyle \left|\mathrm {Cl} {\bigl (}\mathbb {Q} (\tau ){\bigr )}\right|} , the class number of Q ( τ ) {\displaystyle \mathbb {Q} (\tau )} and the minimal (monic integral) polynomial it satisfies is called the 'Hilbert class polynomial'. Thus if the imaginary quadratic extension Q ( τ ) {\displaystyle \mathbb {Q} (\tau )} has class number 1 (so d is a Heegner number), the j-invariant is an integer. The q-expansion of j, with its Fourier series expansion written as a Laurent series in terms of q = e 2 π i τ {\displaystyle q=e^{2\pi i\tau }} , begins as:
j ( τ ) = 1 q + 744 + 196 884 q + ⋯ . {\displaystyle j(\tau )={\frac {1}{q}}+744+196\,884q+\cdots .}
The coefficients c n {\displaystyle c_{n}} asymptotically grow as
ln ( c n ) ∼ 4 π n + O ( ln ( n ) ) , {\displaystyle \ln(c_{n})\sim 4\pi {\sqrt {n}}+O{\bigl (}\ln(n){\bigr )},}
and the low order coefficients grow more slowly than 200 000 n {\displaystyle 200\,000^{n}} , so for q ≪ 1 200 000 {\displaystyle \textstyle q\ll {\frac {1}{200\,000}}} , j is very well approximated by its first two terms. Setting τ = 1 + − 163 2 {\displaystyle \textstyle \tau ={\frac {1+{\sqrt {-163}}}{2}}} yields
q = − e − π 163 ∴ 1 q = − e π 163 . {\displaystyle q=-e^{-\pi {\sqrt {163}}}\quad \therefore \quad {\frac {1}{q}}=-e^{\pi {\sqrt {163}}}.}
Now
j ( 1 + − 163 2 ) = ( − 640 320 ) 3 , {\displaystyle j\left({\frac {1+{\sqrt {-163}}}{2}}\right)=\left(-640\,320\right)^{3},}
so,
( − 640 320 ) 3 = − e π 163 + 744 + O ( e − π 163 ) . {\displaystyle \left(-640\,320\right)^{3}=-e^{\pi {\sqrt {163}}}+744+O\left(e^{-\pi {\sqrt {163}}}\right).}
Or,
e π 163 = 640 320 3 + 744 + O ( e − π 163 ) {\displaystyle e^{\pi {\sqrt {163}}}=640\,320^{3}+744+O\left(e^{-\pi {\sqrt {163}}}\right)}
where the linear term of the error is,
− 196 884 e π 163 ≈ − 196 884 640 320 3 + 744 ≈ − 0.000 000 000 000 75 {\displaystyle {\frac {-196\,884}{e^{\pi {\sqrt {163}}}}}\approx {\frac {-196\,884}{640\,320^{3}+744}}\approx -0.000\,000\,000\,000\,75}
explaining why e π 163 {\displaystyle e^{\pi {\sqrt {163}}}} is within approximately the above of being an integer.
Pi formulas The Chudnovsky brothers found in 1987 that
1 π = 12 640 320 3 2 ∑ k = 0 ∞ ( 6 k ) ! ( 163 ⋅ 3 344 418 k + 13 591 409 ) ( 3 k ) ! ( k ! ) 3 ( − 640 320 ) 3 k , {\displaystyle {\frac {1}{\pi }}={\frac {12}{640\,320^{\frac {3}{2}}}}\sum _{k=0}^{\infty }{\frac {(6k)!(163\cdot 3\,344\,418k+13\,591\,409)}{(3k)!(k!)^{3}(-640\,320)^{3k}}},}
a proof of which uses the fact that
j ( 1 + − 163 2 ) = − 640 320 3 . {\displaystyle j\left({\frac {1+{\sqrt {-163}}}{2}}\right)=-640\,320^{3}.}
For similar formulas, see the Ramanujan–Sato series.
Other Heegner numbers For the four largest Heegner numbers, the approximations one obtains are as follows.
e π 19 ≈ 000 0 96 3 + 744 − 0.22 e π 43 ≈ 000 960 3 + 744 − 0.000 22 e π 67 ≈ 00 5 280 3 + 744 − 0.000 0013 e π 163 ≈ 640 320 3 + 744 − 0.000 000 000 000 75 {\displaystyle {\begin{aligned}e^{\pi {\sqrt {19}}}&\approx {\phantom {000\,0}}96^{3}+744-0.22\\e^{\pi {\sqrt {43}}}&\approx {\phantom {000\,}}960^{3}+744-0.000\,22\\e^{\pi {\sqrt {67}}}&\approx {\phantom {00}}5\,280^{3}+744-0.000\,0013\\e^{\pi {\sqrt {163}}}&\approx 640\,320^{3}+744-0.000\,000\,000\,000\,75\end{aligned}}}
Alternatively,
e π 19 ≈ 12 3 ( 3 2 − 1 ) 3 00 + 744 − 0.22 e π 43 ≈ 12 3 ( 9 2 − 1 ) 3 00 + 744 − 0.000 22 e π 67 ≈ 12 3 ( 21 2 − 1 ) 3 0 + 744 − 0.000 0013 e π 163 ≈ 12 3 ( 231 2 − 1 ) 3 + 744 − 0.000 000 000 000 75 {\displaystyle {\begin{aligned}e^{\pi {\sqrt {19}}}&\approx 12^{3}\left(3^{2}-1\right)^{3}{\phantom {00}}+744-0.22\\e^{\pi {\sqrt {43}}}&\approx 12^{3}\left(9^{2}-1\right)^{3}{\phantom {00}}+744-0.000\,22\\e^{\pi {\sqrt {67}}}&\approx 12^{3}\left(21^{2}-1\right)^{3}{\phantom {0}}+744-0.000\,0013\\e^{\pi {\sqrt {163}}}&\approx 12^{3}\left(231^{2}-1\right)^{3}+744-0.000\,000\,000\,000\,75\end{aligned}}}
where the reason for the squares is due to certain Eisenstein series. For Heegner numbers d < 19 {\displaystyle d<19} , one does not obtain an almost integer; even d = 19 {\displaystyle d=19} is not noteworthy. The integer j-invariants are highly factorisable, which follows from the form
12 3 ( n 2 − 1 ) 3 = ( 2 2 ⋅ 3 ⋅ ( n − 1 ) ⋅ ( n + 1 ) ) 3 , {\displaystyle 12^{3}\left(n^{2}-1\right)^{3}=\left(2^{2}\cdot 3\cdot (n-1)\cdot (n+1)\right)^{3},}
and factor as,
j ( 1 + − 19 2 ) = 000 0 − 96 3 = − ( 2 5 ⋅ 3 ) 3 j ( 1 + − 43 2 ) = 000 − 960 3 = − ( 2 6 ⋅ 3 ⋅ 5 ) 3 j ( 1 + − 67 2 ) = 00 − 5 280 3 = − ( 2 5 ⋅ 3 ⋅ 5 ⋅ 11 ) 3 j ( 1 + − 163 2 ) = − 640 320 3 = − ( 2 6 ⋅ 3 ⋅ 5 ⋅ 23 ⋅ 29 ) 3 . {\displaystyle {\begin{aligned}j\left({\frac {1+{\sqrt {-19}}}{2}}\right)&={\phantom {000\,0}}-96^{3}=-\left(2^{5}\cdot 3\right)^{3}\\j\left({\frac {1+{\sqrt {-43}}}{2}}\right)&={\phantom {000\,}}-960^{3}=-\left(2^{6}\cdot 3\cdot 5\right)^{3}\\j\left({\frac {1+{\sqrt {-67}}}{2}}\right)&={\phantom {00}}-5\,280^{3}=-\left(2^{5}\cdot 3\cdot 5\cdot 11\right)^{3}\\j\left({\frac {1+{\sqrt {-163}}}{2}}\right)&=-640\,320^{3}=-\left(2^{6}\cdot 3\cdot 5\cdot 23\cdot 29\right)^{3}.\end{aligned}}}
These transcendental numbers, in addition to being closely approximated by integers (which are simply algebraic numbers of degree 1), can be closely approximated by algebraic numbers of degree 3,
e π 19 ≈ x 24 − 24.000 31 ; x 3 − 2 x − 2 = 0 e π 43 ≈ x 24 − 24.000 000 31 ; x 3 − 2 x 2 − 2 = 0 e π 67 ≈ x 24 − 24.000 000 0019 ; x 3 − 2 x 2 − 2 x − 2 = 0 e π 163 ≈ x 24 − 24.000 000 000 000 0011 ; x 3 − 6 x 2 + 4 x − 2 = 0 {\displaystyle {\begin{aligned}e^{\pi {\sqrt {19}}}&\approx x^{24}-24.000\,31;&x^{3}-2x-2&=0\\e^{\pi {\sqrt {43}}}&\approx x^{24}-24.000\,000\,31;&x^{3}-2x^{2}-2&=0\\e^{\pi {\sqrt {67}}}&\approx x^{24}-24.000\,000\,0019;&x^{3}-2x^{2}-2x-2&=0\\e^{\pi {\sqrt {163}}}&\approx x^{24}-24.000\,000\,000\,000\,0011;&\quad x^{3}-6x^{2}+4x-2&=0\end{aligned}}}
The roots of the cubics can be exactly given by quotients of the Dedekind eta function η(τ), a modular function involving a 24th root, and which explains the 24 in the approximation. They can also be closely approximated by algebraic numbers of degree 4,
e π 19 ≈ 3 5 ( 3 − 2 ( 1 − 96 24 + 1 3 ⋅ 19 ) ) − 2 − 12.000 06 … e π 43 ≈ 3 5 ( 9 − 2 ( 1 − 960 24 + 7 3 ⋅ 43 ) ) − 2 − 12.000 000 061 … e π 67 ≈ 3 5