The Sine Test

THE PRIME NUMBER SERIES 24

The Sine Test

The sine test is a prime number test that unambiguously determines whether a chosen natural number greater than 2 is a prime number or not.

Let N be an element of the natural numbers greater than 2. Inserted in the following formula, it can be calculated whether N is a prime number or a composite number:

$ts(N)=\sin^2(\frac{N}{2}\pi)\prod\limits_{k=0} ^{k_{max}} \lceil\sin^2(\frac{N}{2k+3}\pi)\rceil$

with

$k_{max}={\frac{\sqrt{N}-3} {2}}$

and k ∈ ℕ and k ≥ 0.

Legend:

$\prod\limits_{k=0} ^{k_{max}}$ product in which the run variable k runs from 0 to ${k_{max}}$

⌈x⌉ upper Gaussian bracket, x rounded up to the nearest whole number

For a chosen N from the set of natural numbers, ts(N) is either 0 or 1.

If ts(N)=0, then N is not prime.

If ts(N)=1, then N is prime.

Examples:

N=3 → ts(3)=1 → 3 is prime

N=4 → ts(4)=0 → 4 is not prime

N=107 → ts(107)=1 → 107 is prime

N=3002 → ts(3002)=0 → 3002 is not prime

N=50993 → ts(50993)=1 → 50993 is prime

N=253997 → ts(253997)=0 → 253997 is not prime

N=7643249 → ts(7643249)=1 → 7643249 is prime

Munich, 7 May 2024

Gottfried Färberböck

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THE PRIME NUMBER SERIES

The Sine Test