15 set

In mathematics , the 15 theorem is a theorem, proven by John Horton Conway and William Schneeberger , about the representability of natural numbers by square forms . He generalizes Lagrange's theorem , according to which every natural number can be decomposed as the sum of four square numbers.

statement

If a positively definite square form has a matrix representation whose entries are all integers , and if the form itself takes all values from 1 to 15, then the form can take all positive integers as values. (The form is then called universal .)

A stronger version of the theorem states that the form is already universal when it takes on the nine values 1, 2, 3, 5, 6, 7, 10, 14 and 15 (sequence A030050 in OEIS ). In this formulation, the sentence is sharp: none of the nine values can be removed from the list, since one can always specify a square form that takes every whole number except for a single number in the list.

history

A forerunner of the 15 theorem is the four-squares theorem , which was suspected by Bachet in 1621 and proven by Lagrange in 1770 .

As a generalization of Lagrange's theorem, Conway and Schneeberger proved the 15-theorem in 1993.

Conway and Schneeberger's evidence was never published. Manjul Bhargava found a simpler proof in 2000 and gave all 204 universal forms.

Examples

The square form produces all integers except the 15. ${\ displaystyle a ^ {2} + 2b ^ {2} + 5c ^ {2} + 5d ^ {2}}$
It's easy to verify that the square shape produces all integers up to and including 15. But Lagrange's theorem does not follow from this, because it is used as an aid in the proof of the 15-theorem and therefore requires an independent proof. ${\ displaystyle a ^ {2} + b ^ {2} + c ^ {2} + d ^ {2}}$

More sentences

Manjul Bhargava also formulated the following sentences analogously:

290 theorem

A positive definite quadratic form is given, which always returns integer values for integer inputs (this is a weaker condition than having an integer matrix). If such a form generates all values up to 290, it also generates all values above 290. The statement can be tightened to a set of 29 numbers that must be generated by the quadratic form, the so-called critical integers:

1, 2, 3, 5, 6, 7, 10, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 34, 35, 37, 42, 58, 93, 110, 145, 203, 290. (Sequence A030051 in OEIS )

33 theorem

A positive definite square form with an integer matrix representation that generates these 7 odd values up to 33 is able to generate all odd numbers.

Set of critical integers: 1, 3, 5, 7, 11, 15, 33. (Sequence A116582 in OEIS )

73 theorem

A positively definite square form with an integer matrix representation that generates these 17 values up to 73 is able to generate all prime numbers .

Set of critical integers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 67, 73. (Sequence A154363 in OEIS )

literature

Marc Chamberland: From one to nine - big miracles behind small numbers. Over 100 mathematical excursions for the curious and connoisseur. Springer-Verlag 2016, ISBN 978-3-662-50250-1
Manjul Bhargava: On the Conway-Schneeberger fifteen theorem. Quadratic forms and their applications (Dublin, 1999), 27-37, Contemp. Math., 272, Amer. Math. Soc., Providence, RI, 2000.
Manjul Bhargava, Jonathan Hanke: Universal quadratic forms and the 290-theorem. Inventiones Mathematicae, 2011 (PDF; 425 KB, 16 pages).