Størmer number


In mathematics, a Størmer number or arc-cotangent irreducible number, named after Carl Størmer, is a positive integer n for which the greatest prime factor of n2 + 1 is greater than or equal to 2n.

Sequence

The first few Størmer numbers are:

Density

proved that this sequence is neither finite nor cofinite.
More precisely, the natural density of the Størmer numbers lies between 0.5324 and 0.905.
It has been conjectured that their natural density is the natural logarithm of 2, approximately 0.693, but this remains unproven.
Because the Størmer numbers have positive density, the Størmer numbers form a large set.

Restrictions

A number of the form 2x2 for x>1 cannot be a Størmer number. This is because 2+1 = 4x4+1 =.

Application

The Størmer numbers arise in connection with the problem of representing the Gregory numbers as sums of Gregory numbers for integers. The Gregory number may be decomposed by repeatedly multiplying the Gaussian integer by numbers of the form, in order to cancel prime factors p from the imaginary part; here is chosen to be a Størmer number such that is divisible by.