Math Is Fun Forum

Stangerzv

Member

Registered: 2012-01-30

Posts: 266

Prime Number with (mod3) I Hope it could be new one.

I have encountered a new property in which I hope nobody has found it yet.

The equation is given as follows:

If p is prime and greater than 3 then,

In other words,

If p is prime then

is a whole number.

Prime generated y is given as follows:

y(7)=- 90825083
y(47)=33333333333333333333333333333315800289254723317
y(79)=3 333333333333333333333333333333333333333333333333333333333330177241305791050933
y(83)=33333333333333333333333333333333333333333333333333333333333333328161319604264715517

Nehushtan

Member

Registered: 2013-03-09

Posts: 957

Re: Prime Number with (mod3) I Hope it could be new one.

This can be easily proved. Since p is not divisible by 3, we have

This comes from the fact that the square of any integer not divisible by 3 is congruent to 1 (mod 3). Also

since 10 ≡ 1 (mod 3). Thus

NB: The result is true if p is any positive integer not divisible by 3.

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Stangerzv

Member

Registered: 2012-01-30

Posts: 266

Re: Prime Number with (mod3) I Hope it could be new one.

Thanks for the proof:) Basically, y also can be prime for odd composite p (i.e. p=299) but I am limiting it only to prime p for making it harder to find.

Last edited by Stangerzv (2013-12-11 00:41:31)