Numerus idoneus

As numerus idoneus ( German  suitable number or suitable number , plural Numbers idonei ) is one of zero various natural number denotes that are not in the form can be represented, with and pairwise different positive integers. Of these numbers 65 are known; the largest of the known numeri idonei is the number 1848. ${\displaystyle ab+ac+bc}$ ${\displaystyle a,}$ ${\displaystyle b}$ ${\displaystyle c}$

Furthermore, it was shown that there can be at most one other suitable number among the square-free numbers , provided that the generalized Riemann Hypothesis (for quadratic characters) is valid.

The 65 known numeri idonei are:

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 15, 16, 18, 21, 22, 24, 25, 28, 30, 33, 37, 40, 42, 45, 48, 57, 58, 60, 70, 72, 78, 85, 88, 93, 102, 105, 112, 120, 130, 133, 165, 168, 177, 190, 210, 232, 240, 253, 273, 280, 312, 330, 345, 357, 385, 408, 462, 520, 760, 840, 1320, 1365 and 1848. (Sequence A000926 in OEIS )

They were all found by Leonhard Euler and Carl Friedrich Gauß .

The designation numerus idoneus goes back to Leonhard Euler, who introduced these numbers in 1778 in connection with the search for large prime numbers , as he assumed that they were "suitable" or "suitable" for a method he had discovered (lat. Idoneus ) are.

Euler's method is based on the following theorem:

Let be a suitable number and be an odd number for which the equation has exactly one integer solution . If and prime , then is prime. ${\displaystyle D}$ ${\displaystyle m}$ ${\displaystyle m=x^{2}+Dy^{2}}$ ${\displaystyle x,y>0}$ ${\displaystyle x}$ ${\displaystyle Dy}$ ${\displaystyle m}$

Within the set of odd numbers in the form , the primality can thus be checked simply by examining coprime and the uniqueness of the representation . ${\displaystyle x^{2}+Dy^{2}}$ ${\displaystyle x^{2}+Dy^{2}}$

literature

• J. Steinig: On Euler's Idoneal Numbers . In: El. Math. Band 22 , no. 4 , 1966, pp. 73-96 (English, online ).
• W. Sierpinski : Elementary Theory of Numbers . Ed .: A. Schinzel. 2nd Edition. Elsevier, 1988, ISBN 0-08-096019-7 , pp. 228/229 (English).
• Paulo Ribenboim: My numbers, my friends: highlights of number theory . Springer-Verlag, 2009, ISBN 978-3-540-87957-2 , p. 373 .
• Günther Frei : Euler's convenient numbers. Mathematical Intelligencer, Volume 7, No. 3, 1985, pp. 55-58.
• Günther Frei: Les nombres convenables de Leonhard Euler. Publ. Math. Fac. Sci. Besançon, Théorie des Nombres, 1983-1984 (1985), pp. 1-58.
• Harald Scheid : Number Theory . 3. Edition. Spectrum Academic Publishing House, Heidelberg (among others) 2003, ISBN 3-8274-1365-6 .

Web links

• Eric W. Weisstein : Ideal Number . In: MathWorld (English).

Individual evidence

1. ↑ Harald Scheid: Number theory. 2003, p. 219.
2. ^ P. Weinberger: Exponents of the class groups of complex quadratic fields . In: Acta Arith, Volume 22 , 1973, p. 117–124 ( online [PDF; 360 kB ]).
3. ↑ Ernst Kani: Idoneal Numbers and some Generalizations . In: Ann. Sci. Math. Québec . tape 35 , no. 2 , 2011, p. 197–227 , p. 12 ( online [PDF; 260 kB ] Corollary 23).
4. ↑ An English translation (from Latin) of the work first presented by Euler in 1778 (and published in 1806) is:
   Leonhard Euler: An illustration of a paradox about the idoneal, or suitable, numbers . In: Nova Acta Academiae Scientarum Imperialis Petropolitinae . tape 15 , 1806, pp. 29-32 , arxiv : math / 0507352 (see also: Leonhard Euler: Opera Omnia. Series 1: Opera mathematica. Volume 4, Birkhäuser, 1992).
5. ↑ J. Steinig: On Euler's Idoneal Numbers . In: El. Math. Band 22 , no. 4 , 1966, pp. 73-96 ( online ).