Inmathematics, aStørmer number orarc-cotangent irreducible number is a positiveinteger${\displaystyle n}$ for which the greatestprime factor of${\displaystyle n^2+1}$ is greater than or equal to${\displaystyle 2n}$. They are named afterCarl Størmer.

The first Størmer numbers below 100 are:

The only numbers below 100 thataren't Størmer are3,7,8,13,17,18,21,30,31,32,38,41,43,46,47,50,55,57,68,70,72,73,75,76,83,91,93,98,99 and100.

John Todd proved that this sequence is neitherfinite norcofinite.^[1]

More precisely, thenatural density of the Størmer numbers lies between 0.5324 and 0.905.It has been conjectured that their natural density is thenatural logarithm of 2, approximately 0.693, but this remains unproven.^[2]Because the Størmer numbers have positive density, the Størmer numbers form alarge set.

The Størmer numbers arise in connection with the problem of representing theGregory numbers (arctangents ofrational numbers)${\displaystyle G_{a/b}=\arctan \frac{b}{a}}$ as sums of Gregory numbers for integers (arctangents ofunit fractions). The Gregory number${\displaystyle G_{a/b}}$ may be decomposed by repeatedly multiplying theGaussian integer${\displaystyle a+bi}$ by numbers of the form${\displaystyle n\pm i}$, in order to cancel prime factors${\displaystyle p}$ from the imaginary part; here${\displaystyle n}$ is chosen to be a Størmer number such that${\displaystyle n^2+1}$ is divisible by${\displaystyle p}$.^[3]

• Integer sequences

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