In mathematics,
Euler's idoneal numbers (also called suitable numbers or convenient numbers) are the positive integers ''D'' such that any integer expressible in only one way as ''x''
2 ± ''Dy''
2 (where ''x''
2 is
relatively prime to ''Dy''
2) is a prime power or twice a prime power. In particular, a number that has two distinct representations as a sum of two squares is
composite
Composite or compositing may refer to:
Materials
* Composite material, a material that is made from several different substances
** Metal matrix composite, composed of metal and other parts
** Cermet, a composite of ceramic and metallic materials
...
. Every idoneal number generates a set containing infinitely many primes and missing infinitely many other primes.
Definition
A positive integer ''n'' is idoneal if and only if it cannot be written as ''ab'' + ''bc'' + ''ac'' for distinct positive integers ''a, b'', and ''c''.
It is sufficient to consider the set ; if all these numbers are of the form , , or ''2''
s for some integer
s, where is a prime, then is idoneal.
Conjecturally complete listing
The 65 idoneal numbers found by
Leonhard Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
and
Carl Friedrich Gauss and conjectured to be the only such numbers 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 .
Results of
Peter J. Weinberger
Peter Jay Weinberger (born August 6, 1942) is a computer scientist best known for his early work at Bell Labs. He now works at Google.
Weinberger was an undergraduate at Swarthmore College, graduating in 1964. He received his PhD in mathemati ...
from 1973 imply that at most two other idoneal numbers exist, and that the list above is complete if the
generalized Riemann hypothesis holds (some sources incorrectly claim that Weinberger's results imply that there's at most one other idoneal number).
See also
*
List of unsolved problems in mathematics
Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Eucli ...
Notes
References
* Z. I. Borevich and I. R. Shafarevich, ''Number Theory''. Academic Press, NY, 1966, pp. 425–430.
*
* L. Euler,
An illustration of a paradox about the idoneal, or suitable, numbers, 1806
* G. Frei, Euler's convenient numbers, Math. Intell. Vol. 7 No. 3 (1985), 55–58 and 64.
* O-H. Keller, Ueber die "Numeri idonei" von Euler, Beitraege Algebra Geom., 16 (1983), 79–91.
ath. Rev. 85m:11019* G. B. Mathews, ''Theory of Numbers'', Chelsea, no date, p. 263.
*
P. Ribenboim, "Galimatias Arithmeticae", in Mathematics Magazine 71(5) 339 1998 MAA or, 'My Numbers, My Friends', Chap.11 Springer-Verlag 2000 NY
* J. Steinig, On Euler's ideoneal numbers, Elemente Math., 21 (1966), 73–88.
*
A. Weil, ''
Number theory: an approach through history; from Hammurapi to Legendre'', Birkhaeuser, Boston, 1984; see p. 188.
* P. Weinberger, Exponents of the class groups of complex quadratic fields, Acta Arith., 22 (1973), 117–124.
* Ernst Kani, Idoneal Numbers And Some Generalizations, Ann. Sci. Math. Québec 35, No 2, (2011), 197-227.
External links
* K. S. Brown, Mathpages
Numeri Idonei* M. Waldschmidt
Open Diophantine problems*
{{Classes of natural numbers
Integer sequences
Unsolved problems in number theory
Leonhard Euler