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Heegner number

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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) has class number 1. Equivalently, its ring of integers has unique factorization.

Contents

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:

1, 2, 3, 7, 11, 19, 43, 67, 163. (sequence A003173 in the OEIS)

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 the gap in Heegner's proof was minor.

Euler's prime-generating polynomial

Euler's prime-generating polynomial

n 2 n + 41 ,

which gives (distinct) primes for n = 1, ..., 40, is related to the Heegner number 163 = 4 · 41 − 1.

Euler's formula, with n taking the values 1,... 40 is equivalent to

n 2 + n + 41 ,

with n taking the values 0,... 39, and Rabinowitz proved that

n 2 + n + p

gives primes for n = 0 , , p 2 if and only if its discriminant 1 4 p equals minus a Heegner number.

(Note that p 1 yields p 2 , so 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 , which is an almost integer, in that it is very close to an integer:

e π 163 = 262 537 412 640 768 743.999 999 999 999 25 640 320 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.

This coincidence is explained by complex multiplication and the q-expansion of the j-invariant.

Detail

Briefly, j ( ( 1 + d ) / 2 ) is an integer for d a Heegner number, and e π d j ( ( 1 + d ) / 2 ) + 744 via the q-expansion.

If τ is a quadratic irrational, then the j-invariant is an algebraic integer of degree | Cl ( Q ( τ ) ) | , the class number of Q ( τ ) and the minimal (monic integral) polynomial it satisfies is called the Hilbert class polynomial. Thus if the imaginary quadratic extension Q ( τ ) 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 = exp ( 2 π i τ ) , begins as:

j ( q ) = 1 q + 744 + 196 884 q + .

The coefficients c n asymptotically grow as ln ( c n ) 4 π n + O ( ln ( n ) ) , and the low order coefficients grow more slowly than 200 000 n , so for q 1 / 200 000 , j is very well approximated by its first two terms. Setting τ = ( 1 + 163 ) / 2 yields q = exp ( π 163 ) or equivalently, 1 q = exp ( π 163 ) . Now j ( ( 1 + 163 ) / 2 ) = ( 640 320 ) 3 , so,

( 640 320 ) 3 = e π 163 + 744 + O ( e π 163 ) .

Or,

e π 163 = 640 320 3 + 744 + O ( e π 163 )

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

explaining why e π 163 is within approximately the above of being an integer.

Pi formulas

The Chudnovsky brothers found in 1987,

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

and uses the fact that j ( 1 + 163 2 ) = 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 96 3 + 744 0.22 e π 43 960 3 + 744 0.000 22 e π 67 5 280 3 + 744 0.000 0013 e π 163 640 320 3 + 744 0.000 000 000 000 75

Alternatively,

e π 19 12 3 ( 3 2 1 ) 3 + 744 0.22 e π 43 12 3 ( 9 2 1 ) 3 + 744 0.000 22 e π 67 12 3 ( 21 2 1 ) 3 + 744 0.000 0013 e π 163 12 3 ( 231 2 1 ) 3 + 744 0.000 000 000 000 75

where the reason for the squares is due to certain Eisenstein series. For Heegner numbers d < 19 , one does not obtain an almost integer; even d = 19 is not noteworthy. The integer j-invariants are highly factorisable, which follows from the 12 3 ( n 2 1 ) 3 = ( 2 2 3 ( n 1 ) ( n + 1 ) ) 3 form, and factor as,

j ( ( 1 + 19 ) / 2 ) = 96 3 = ( 2 5 3 ) 3 j ( ( 1 + 43 ) / 2 ) = 960 3 = ( 2 6 3 5 ) 3 j ( ( 1 + 67 ) / 2 ) = 5 280 3 = ( 2 5 3 5 11 ) 3 j ( ( 1 + 163 ) / 2 ) = 640 320 3 = ( 2 6 3 5 23 29 ) 3 .

These transcendental numbers, in addition to being closely approximated by integers, (which are simply algebraic numbers of degree 1), can also be closely approximated by algebraic numbers of degree 3,

e π 19 x 24 24 ; x 3 2 x 2 = 0 e π 43 x 24 24 ; x 3 2 x 2 2 = 0 e π 67 x 24 24 ; x 3 2 x 2 2 x 2 = 0 e π 163 x 24 24 ; x 3 6 x 2 + 4 x 2 = 0

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. In addition, they can also be closely approximated by algebraic numbers of degree 4,

e π 19 3 5 ( 3 2 ( 3 + 1 3 19 ) ) 2 12.000 06 e π 43 3 5 ( 9 2 ( 39 + 7 3 43 ) ) 2 12.000 000 061 e π 67 3 5 ( 21 2 ( 219 + 31 3 67 ) ) 2 12.000 000 000 36 e π 163 3 5 ( 231 2 ( 26 679 + 2 413 3 163 ) ) 2 12.000 000 000 000 000 21

Note the reappearance of the integers n = 3 , 9 , 21 , 231 as well as the fact that,

2 6 3 ( 3 2 + 3 19 1 2 ) = 96 2 2 6 3 ( 39 2 + 3 43 7 2 ) = 960 2 2 6 3 ( 219 2 + 3 67 31 2 ) = 5 280 2 2 6 3 ( 26679 2 + 3 163 2413 2 ) = 640 320 2

which, with the appropriate fractional power, are precisely the j-invariants. As well as 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

where the xs are given respectively by the appropriate root of the sextic equations,

5 x 6 96 x 5 10 x 3 + 1 = 0 5 x 6 960 x 5 10 x 3 + 1 = 0 5 x 6 5 280 x 5 10 x 3 + 1 = 0 5 x 6 640 320 x 5 10 x 3 + 1 = 0

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 (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 , 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

where the eta quotients are the algebraic numbers given above.

Consecutive primes

Given an odd prime p, if one computes k 2 ( mod p ) for k = 0 , 1 , , ( p 1 ) / 2 (this is sufficient because ( p k ) 2 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.

References

Heegner number Wikipedia


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