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In number theory, a Heegner number (as termed by Conway and Guy) is a square-free positive integer such that the imaginary quadratic field has class number . Equivalently, its ring of integers has unique factorization.[1]

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:

. (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.[2]

Contents

Euler's prime-generating polynomialEdit

Euler's prime-generating polynomial

 

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

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

 

with   taking the values 0,... 39, and Rabinowitz[3] proved that

 

gives primes for   if and only if its discriminant   is the negative of a Heegner number.

(Note that   yields  , so   is maximal.) 1, 2, and 3 are not of the required form, so the Heegner numbers that work are  , yielding prime generating functions of Euler's form for  ; these latter numbers are called lucky numbers of Euler by F. Le Lionnais.[4]

Almost integers and Ramanujan's constantEdit

Ramanujan's constant is the transcendental number[5] , which is an almost integer, in that it is very close to an integer:

 [6]  

This number was discovered in 1859 by the mathematician Charles Hermite.[7] In a 1975 April Fool article in Scientific American magazine,[8] "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.

DetailEdit

Briefly,   is an integer for d a Heegner number, and   via the q-expansion.

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

 

The coefficients   asymptotically grow as  , and the low order coefficients grow more slowly than  , so for  , j is very well approximated by its first two terms. Setting   yields   or equivalently,  . Now  , so,

 

Or,

 

where the linear term of the error is,

 

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

Pi formulasEdit

The Chudnovsky brothers found in 1987 that

 

which uses the fact that  . For similar formulas, see the Ramanujan–Sato series.

Other Heegner numbersEdit

For the four largest Heegner numbers, the approximations one obtains[9] are as follows.

 

Alternatively,[10]

 

where the reason for the squares is due to certain Eisenstein series. For Heegner numbers  , one does not obtain an almost integer; even   is not noteworthy.[11] The integer j-invariants are highly factorisable, which follows from the   form, and factor as,

 

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,[12]

 

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,[13]

 

If   denotes the expression in the parenthesis (e.g.  ), it satisfies respectively the quartic equations

 

Note the reappearance of the integers   as well as the fact that

 

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

Similarly for algebraic numbers of degree 6,

 

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

 

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   (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  , then,

 

where the eta quotients are the algebraic numbers given above.

Class 2 numbersEdit

The three numbers  , for which the imaginary quadratic field   has class number  , are not considered as Heegner numbers but have certain similar properties in terms of almost integers. For instance, we have

 

and

 

Consecutive primesEdit

Given an odd prime p, if one computes   for   (this is sufficient because  ), one gets consecutive composites, followed by consecutive primes, if and only if p is a Heegner number.[14]

For details, see "Quadratic Polynomials Producing Consecutive Distinct Primes and Class Groups of Complex Quadratic Fields" by Richard Mollin.[15]

Notes and referencesEdit

  1. ^ Conway, John Horton; Guy, Richard K. (1996). The Book of Numbers. Springer. p. 224. ISBN 0-387-97993-X.
  2. ^ Stark, H. M. (1969), "On the gap in the theorem of Heegner" (PDF), Journal of Number Theory, 1: 16–27, doi:10.1016/0022-314X(69)90023-7
  3. ^ Rabinovitch, Georg "Eindeutigkeit der Zerlegung in Primzahlfaktoren in quadratischen Zahlkörpern." Proc. Fifth Internat. Congress Math. ( Cambridge) 1, 418–421, 1913.
  4. ^ Le Lionnais, F. Les nombres remarquables. Paris: Hermann, pp. 88 and 144, 1983.
  5. ^ Weisstein, Eric W. "Transcendental Number". MathWorld. gives  , based on Nesterenko, Yu. V. "On Algebraic Independence of the Components of Solutions of a System of Linear Differential Equations." Izv. Akad. Nauk SSSR, Ser. Mat. 38, 495–512, 1974. English translation in Math. USSR 8, 501–518, 1974.
  6. ^ Ramanujan Constant – from Wolfram MathWorld
  7. ^ Barrow, John D (2002). The Constants of Nature. London: Jonathan Cape. ISBN 0-224-06135-6.
  8. ^ Gardner, Martin (April 1975). "Mathematical Games". Scientific American. Scientific American, Inc. 232 (4): 127.
  9. ^ These can be checked by computing   on a calculator, and   for the linear term of the error.
  10. ^ http://groups.google.com.ph/group/sci.math.research/browse_thread/thread/3d24137c9a860893?hl=en#
  11. ^ The absolute deviation of a random real number (picked uniformly from [0,1], say) is a uniformly distributed variable on [0, 0.5], so it has absolute average deviation and median absolute deviation of 0.25, and a deviation of 0.22 is not exceptional.
  12. ^ "Pi Formulas".
  13. ^ "Extending Ramanujan's Dedekind Eta Quotients".
  14. ^ http://www.mathpages.com/home/kmath263.htm
  15. ^ Mollin, R. A. (1996). "Quadratic polynomials producing consecutive, distinct primes and class groups of complex quadratic fields" (PDF). Acta Arithmetica. 74: 17–30.

External linksEdit