- Source: Heegner number
In number theory, a Heegner number (as termed by Conway and Guy) is a square-free positive integer d such that the imaginary quadratic field
Q
[
−
d
]
{\displaystyle \mathbb {Q} \left[{\sqrt {-d}}\right]}
has class number 1. Equivalently, the ring of algebraic integers of
Q
[
−
d
]
{\displaystyle \mathbb {Q} \left[{\sqrt {-d}}\right]}
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 (Baker–)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, is related to the Heegner number 163 = 4 · 41 − 1.
Rabinowitz proved that
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 this quadratic's discriminant
1
−
4
p
{\displaystyle 1-4p}
is the negative of a Heegner number.
(Note that
p
−
1
{\displaystyle p-1}
yields
p
2
{\displaystyle p^{2}}
, so
p
−
2
{\displaystyle p-2}
is maximal.)
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 (de facto it is one):
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 Fool 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. In this wise it has as a spurious provenance as the Feynman point.
This coincidence is explained by complex multiplication and the q-expansion of the j-invariant.
= Detail
=In what follows, j(z) denotes the j-invariant of the complex number 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 (}\mathbf {Q} (\tau ){\bigr )}\right|}
, the class number of
Q
(
τ
)
{\displaystyle \mathbf {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 \mathbf {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 {\color {white}000\,0}96^{3}+744-0.22\\e^{\pi {\sqrt {43}}}&\approx {\color {white}000\,}960^{3}+744-0.000\,22\\e^{\pi {\sqrt {67}}}&\approx {\color {white}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}{\color {white}00}+744-0.22\\e^{\pi {\sqrt {43}}}&\approx 12^{3}\left(9^{2}-1\right)^{3}{\color {white}00}+744-0.000\,22\\e^{\pi {\sqrt {67}}}&\approx 12^{3}\left(21^{2}-1\right)^{3}{\color {white}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)&={\color {white}000\,0}-96^{3}=-\left(2^{5}\cdot 3\right)^{3}\\j\left({\frac {1+{\sqrt {-43}}}{2}}\right)&={\color {white}000\,}-960^{3}=-\left(2^{6}\cdot 3\cdot 5\right)^{3}\\j\left({\frac {1+{\sqrt {-67}}}{2}}\right)&={\color {white}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
(
21
−
2
(
1
−
5
280
24
+
31
3
⋅
67
)
)
−
2
−
12.000
000
000
36
…
e
π
163
≈
3
5
(
231
−
2
(
1
−
640
320
24
+
2
413
3
⋅
163
)
)
−
2
−
12.000
000
000
000
000
21
…
{\displaystyle {\begin{aligned}e^{\pi {\sqrt {19}}}&\approx 3^{5}\left(3-{\sqrt {2\left(1-{\tfrac {96}{24}}+1{\sqrt {3\cdot 19}}\right)}}\right)^{-2}-12.000\,06\dots \\e^{\pi {\sqrt {43}}}&\approx 3^{5}\left(9-{\sqrt {2\left(1-{\tfrac {960}{24}}+7{\sqrt {3\cdot 43}}\right)}}\right)^{-2}-12.000\,000\,061\dots \\e^{\pi {\sqrt {67}}}&\approx 3^{5}\left(21-{\sqrt {2\left(1-{\tfrac {5\,280}{24}}+31{\sqrt {3\cdot 67}}\right)}}\right)^{-2}-12.000\,000\,000\,36\dots \\e^{\pi {\sqrt {163}}}&\approx 3^{5}\left(231-{\sqrt {2\left(1-{\tfrac {640\,320}{24}}+2\,413{\sqrt {3\cdot 163}}\right)}}\right)^{-2}-12.000\,000\,000\,000\,000\,21\dots \end{aligned}}}
If
x
{\displaystyle x}
denotes the expression within the parenthesis (e.g.
x
=
3
−
2
(
1
−
96
24
+
1
3
⋅
19
)
{\displaystyle x=3-{\sqrt {2\left(1-{\tfrac {96}{24}}+1{\sqrt {3\cdot 19}}\right)}}}
), it satisfies respectively the quartic equations
x
4
−
00
4
⋅
3
x
3
+
000
0
2
3
(
96
+
3
)
x
2
−
000
000
2
3
⋅
3
(
96
−
6
)
x
−
3
=
0
x
4
−
00
4
⋅
9
x
3
+
000
2
3
(
960
+
3
)
x
2
−
000
00
2
3
⋅
9
(
960
−
6
)
x
−
3
=
0
x
4
−
0
4
⋅
21
x
3
+
00
2
3
(
5
280
+
3
)
x
2
−
000
2
3
⋅
21
(
5
280
−
6
)
x
−
3
=
0
x
4
−
4
⋅
231
x
3
+
2
3
(
640
320
+
3
)
x
2
−
2
3
⋅
231
(
640
320
−
6
)
x
−
3
=
0
{\displaystyle {\begin{aligned}x^{4}-{\color {white}00}4\cdot 3x^{3}+{\color {white}000\,0}{\tfrac {2}{3}}(96+3)x^{2}-{\color {white}000\,000}{\tfrac {2}{3}}\cdot 3(96-6)x-3&=0\\x^{4}-{\color {white}00}4\cdot 9x^{3}+{\color {white}000\,}{\tfrac {2}{3}}(960+3)x^{2}-{\color {white}000\,00}{\tfrac {2}{3}}\cdot 9(960-6)x-3&=0\\x^{4}-{\color {white}0}4\cdot 21x^{3}+{\color {white}00}{\tfrac {2}{3}}(5\,280+3)x^{2}-{\color {white}000}{\tfrac {2}{3}}\cdot 21(5\,280-6)x-3&=0\\x^{4}-4\cdot 231x^{3}+{\tfrac {2}{3}}(640\,320+3)x^{2}-{\tfrac {2}{3}}\cdot 231(640\,320-6)x-3&=0\\\end{aligned}}}
Note the reappearance of the integers
n
=
3
,
9
,
21
,
231
{\displaystyle n=3,9,21,231}
as well as the fact that
2
6
⋅
3
(
−
(
1
−
96
24
)
2
+
1
2
⋅
3
⋅
19
)
=
96
2
2
6
⋅
3
(
−
(
1
−
960
24
)
2
+
7
2
⋅
3
⋅
43
)
=
960
2
2
6
⋅
3
(
−
(
1
−
5
280
24
)
2
+
31
2
⋅
3
⋅
67
)
=
5
280
2
2
6
⋅
3
(
−
(
1
−
640
320
24
)
2
+
2413
2
⋅
3
⋅
163
)
=
640
320
2
{\displaystyle {\begin{aligned}2^{6}\cdot 3\left(-\left(1-{\tfrac {96}{24}}\right)^{2}+1^{2}\cdot 3\cdot 19\right)&=96^{2}\\2^{6}\cdot 3\left(-\left(1-{\tfrac {960}{24}}\right)^{2}+7^{2}\cdot 3\cdot 43\right)&=960^{2}\\2^{6}\cdot 3\left(-\left(1-{\tfrac {5\,280}{24}}\right)^{2}+31^{2}\cdot 3\cdot 67\right)&=5\,280^{2}\\2^{6}\cdot 3\left(-\left(1-{\tfrac {640\,320}{24}}\right)^{2}+2413^{2}\cdot 3\cdot 163\right)&=640\,320^{2}\end{aligned}}}
which, with the appropriate fractional power, are precisely the j-invariants.
Similarly for algebraic numbers of degree 6,
e
π
19
≈
(
5
x
)
3
−
6.000
010
…
e
π
43
≈
(
5
x
)
3
−
6.000
000
010
…
e
π
67
≈
(
5
x
)
3
−
6.000
000
000
061
…
e
π
163
≈
(
5
x
)
3
−
6.000
000
000
000
000
034
…
{\displaystyle {\begin{aligned}e^{\pi {\sqrt {19}}}&\approx \left(5x\right)^{3}-6.000\,010\dots \\e^{\pi {\sqrt {43}}}&\approx \left(5x\right)^{3}-6.000\,000\,010\dots \\e^{\pi {\sqrt {67}}}&\approx \left(5x\right)^{3}-6.000\,000\,000\,061\dots \\e^{\pi {\sqrt {163}}}&\approx \left(5x\right)^{3}-6.000\,000\,000\,000\,000\,034\dots \end{aligned}}}
where the xs are given respectively by the appropriate root of the sextic equations,
5
x
6
−
000
0
96
x
5
−
10
x
3
+
1
=
0
5
x
6
−
000
960
x
5
−
10
x
3
+
1
=
0
5
x
6
−
00
5
280
x
5
−
10
x
3
+
1
=
0
5
x
6
−
640
320
x
5
−
10
x
3
+
1
=
0
{\displaystyle {\begin{aligned}5x^{6}-{\color {white}000\,0}96x^{5}-10x^{3}+1&=0\\5x^{6}-{\color {white}000\,}960x^{5}-10x^{3}+1&=0\\5x^{6}-{\color {white}00}5\,280x^{5}-10x^{3}+1&=0\\5x^{6}-640\,320x^{5}-10x^{3}+1&=0\end{aligned}}}
with the j-invariants appearing again. These sextics are not only algebraic, they are also solvable in radicals as they factor into two cubics over the extension
Q
5
{\displaystyle \mathbb {Q} {\sqrt {5}}}
(with the first factoring further into two quadratics). These algebraic approximations can be exactly expressed in terms of Dedekind eta quotients. As an example, let
τ
=
1
+
−
163
2
{\displaystyle \textstyle \tau ={\frac {1+{\sqrt {-163}}}{2}}}
, then,
e
π
163
=
(
e
π
i
24
η
(
τ
)
η
(
2
τ
)
)
24
−
24.000
000
000
000
001
05
…
e
π
163
=
(
e
π
i
12
η
(
τ
)
η
(
3
τ
)
)
12
−
12.000
000
000
000
000
21
…
e
π
163
=
(
e
π
i
6
η
(
τ
)
η
(
5
τ
)
)
6
−
6.000
000
000
000
000
034
…
{\displaystyle {\begin{aligned}e^{\pi {\sqrt {163}}}&=\left({\frac {e^{\frac {\pi i}{24}}\eta (\tau )}{\eta (2\tau )}}\right)^{24}-24.000\,000\,000\,000\,001\,05\dots \\e^{\pi {\sqrt {163}}}&=\left({\frac {e^{\frac {\pi i}{12}}\eta (\tau )}{\eta (3\tau )}}\right)^{12}-12.000\,000\,000\,000\,000\,21\dots \\e^{\pi {\sqrt {163}}}&=\left({\frac {e^{\frac {\pi i}{6}}\eta (\tau )}{\eta (5\tau )}}\right)^{6}-6.000\,000\,000\,000\,000\,034\dots \end{aligned}}}
where the eta quotients are the algebraic numbers given above.
Class 2 numbers
The three numbers 88, 148, 232, for which the imaginary quadratic field
Q
[
−
d
]
{\displaystyle \mathbb {Q} \left[{\sqrt {-d}}\right]}
has class number 2, are not Heegner numbers but have certain similar properties in terms of almost integers. For instance,
e
π
88
+
8
744
≈
00
00
2
508
952
2
−
0.077
…
e
π
148
+
8
744
≈
00
199
148
648
2
−
0.000
97
…
e
π
232
+
8
744
≈
24
591
257
752
2
−
0.000
0078
…
{\displaystyle {\begin{aligned}e^{\pi {\sqrt {88}}}+8\,744&\approx {\color {white}00\,00}2\,508\,952^{2}-0.077\dots \\e^{\pi {\sqrt {148}}}+8\,744&\approx {\color {white}00\,}199\,148\,648^{2}-0.000\,97\dots \\e^{\pi {\sqrt {232}}}+8\,744&\approx 24\,591\,257\,752^{2}-0.000\,0078\dots \\\end{aligned}}}
and
e
π
22
−
24
≈
00
(
6
+
4
2
)
6
+
0.000
11
…
e
π
37
+
24
≈
(
12
+
2
37
)
6
−
0.000
0014
…
e
π
58
−
24
≈
(
27
+
5
29
)
6
−
0.000
000
0011
…
{\displaystyle {\begin{aligned}e^{\pi {\sqrt {22}}}-24&\approx {\color {white}00}\left(6+4{\sqrt {2}}\right)^{6}+0.000\,11\dots \\e^{\pi {\sqrt {37}}}+24&\approx \left(12+2{\sqrt {37}}\right)^{6}-0.000\,0014\dots \\e^{\pi {\sqrt {58}}}-24&\approx \left(27+5{\sqrt {29}}\right)^{6}-0.000\,000\,0011\dots \\\end{aligned}}}
Consecutive primes
Given an odd prime p, if one computes
k
2
mod
p
{\displaystyle k^{2}\mod p}
for
k
=
0
,
1
,
…
,
p
−
1
2
{\displaystyle \textstyle k=0,1,\dots ,{\frac {p-1}{2}}}
(this is sufficient because
(
p
−
k
)
2
≡
k
2
mod
p
{\displaystyle \left(p-k\right)^{2}\equiv k^{2}\mod p}
), one gets consecutive composites, followed by consecutive primes, if and only if p is a Heegner number.
For details, see "Quadratic Polynomials Producing Consecutive Distinct Primes and Class Groups of Complex Quadratic Fields" by Richard Mollin.
Notes and references
External links
Weisstein, Eric W. "Heegner Number". MathWorld.
OEIS sequence A003173 (Heegner numbers: imaginary quadratic fields with unique factorization)
Gauss' Class Number Problem for Imaginary Quadratic Fields, by Dorian Goldfeld: Detailed history of problem.
Clark, Alex. "163 and Ramanujan Constant". Numberphile. Brady Haran. Archived from the original on 2013-05-16. Retrieved 2013-04-02.
Kata Kunci Pencarian:
- Heegner number
- 19 (number)
- Kurt Heegner
- 67 (number)
- Stark–Heegner theorem
- 43 (number)
- Class number problem
- Heegner
- 21 (number)
- Quadratic field
