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• ## Trigonometric functions with prime numbers?

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• I have been fooling around with some trigonometric functions and I have noticed an easy primality test with them. One simply plugs in a number p into the
Message 1 of 3 , May 5, 2001
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I have been fooling around with some trigonometric
functions and I have noticed an easy primality test
with them. One simply plugs in a number p into the
equation, and if the answer is an integer, than the
number p is prime! Have there been any recent or
non-recent discoveries relating trigonometic functions
to prime numbers?

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• ... If you permit cosh to be considered trigonometric (it can be reformulated in terms of sin and cos with splashings of sqrt(-1)), then according to
Message 1 of 3 , May 5, 2001
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On Sat, 05 May 2001, Anonymous Anonymous wrote:
> I have been fooling around with some trigonometric
> functions and I have noticed an easy primality test
> with them. One simply plugs in a number p into the
> equation, and if the answer is an integer, than the
> number p is prime! Have there been any recent or
> non-recent discoveries relating trigonometic functions
> to prime numbers?

If you permit 'cosh' to be considered trigonometric (it can be reformulated in terms of sin and cos with splashings of sqrt(-1)), then according to http://www.utm.edu/research/primes/prove/prove3_2.html

<<<
Joerg Arndt notes that a striking (but computationally useless) way to state this test is as follows:

Theorem: p=2^n-1 is prime if and only if p divides cosh(2^(n-2)log(2+sqrt(3))).
>>>

Phil

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• ... Here s an extremely useless, yet mathematically correct trig test: C(n)=(cos(pi*((n-1)!+1)/n))^2 For n 1, C(n)=1 if and only if n is prime :-) Ribenboim
Message 1 of 3 , May 5, 2001
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progboy1 wrote:

> I have been fooling around with some trigonometric functions

Here's an extremely useless, yet mathematically correct
trig test:

C(n)=(cos(pi*((n-1)!+1)/n))^2

For n>1, C(n)=1 if and only if n is prime :-)

Ribenboim credits this piece of nonsense to someone
called Willans.

David
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