Are There Any Other General Forms of Prime Numbers Besides 6n+/-5 and 6n+/-1?

[chhitiz]

is there any other general form of prime no.s known except 6n+/-5 and 6n+/-1. is there any general form of n such that 6n+/-5 or 6n+/-1 is a composite no.?

 

[matticus]

4n+/-1
6n+/-1 is composite iff there are nonzero integers a and b such that n = 6ab + a + b.
for instance 6(4) + 1 is composite since 4 = 6(-1)(-1) + (-1) + (- 1)

 

[CRGreathouse]

is there any other general form of prime no.s known except 6n+/-5 and 6n+/-1.

As many as you'd like. 2n + 1, for example.

is there any general form of n such that 6n+/-5 or 6n+/-1 is a composite no.?

n = 141.

n = 5k.

n in {1, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 19, 20, 21, 22, 23, 24, 25, ...}

n in {k | k - 5, k - 1, k + 1, or k + 5 can be written as ab with 1 < a <= b}

Assuming, of course, that we interpret your statement identically.

 

[chhitiz]

n in {1, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 19, 20, 21, 22, 23, 24, 25, ...}

.

but, then, if i am not wrong, this series doesn't have a pattern, does it?
what i meant by my question was if there is a general form or, a set of general forms which can represent each and every prime no. exhaustively.

 

[CRGreathouse]

but, then, if i am not wrong, this series doesn't have a pattern, does it?

Wow, impredicativity in real life!

The sequence has a pattern, it's stated just below it.

 

[chhitiz]

n in {k | k - 5, k - 1, k + 1, or k + 5 can be written as ab with 1 < a <= b}
i should've asked this earlier, i have no idea what that line between n and k stands for. so i can't understand what this statement means.

 

[CRGreathouse]

i have no idea what that line between n and k stands for.

"such that"

 

[chhitiz]

oh now i get it.but that just comes directly from the definition of prime no.s.
let me rephrase my original question- is there any general form/set of forms an+/-b which expresses every prime no. exhaustively, barring the cases where n=ck+/-d where a,b,c,d are constants and n,k integers>=0. i repeat again, this set of forms should represent every prime no. exhaustively.

 

[CRGreathouse]

let me rephrase my original question- is there any general form/set of forms an+/-b which expresses every prime no. exhaustively, barring the cases where n=ck+/-d where a,b,c,d are constants and n,k integers>=0. i repeat again, this set of forms should represent every prime no. exhaustively.

Sure, r for r a real number. Also r^2 + pi/2 (but not r^2 + pi). Also a^2 + b^2 + c^2 + d^2 for a, b, c, and d integers.

 

[chhitiz]

a^2 + b^2 + c^2 + d^2 for a, b, c, and d integers.
but then if a=b=c=d=1, we get 4. how is this prime? and, is it r^(2+pi/2) or r^2+pi/2

 

[CRGreathouse]

a^2 + b^2 + c^2 + d^2 for a, b, c, and d integers.
but then if a=b=c=d=1, we get 4. how is this prime?

6n+1 for n = 4, how is this prime?

You asked for forms that cover all the primes, not for forms that were only prime.

and, is it r^(2+pi/2) or r^2+pi/2

I intended the second, but both work.

 

[matt grime]

There is no choice of a and b so that an+b is prime for every n.

There is no choice of a and b (except for the trivial a=1 b=0 type) so that every prime is of the form an+b for some n (above you'll note that you have a collection of choices that will give every prime, with exceptions such as 2 and 3 for the 6n+1 and 6n-1 case).

The 'pattern' of the primes is entirely deterministic (sieve of what's-his-face) and simultaneously very hard to prove anything about (e.g. twin prime conjecture).

At least that is what I think you're getting at.

 

[chhitiz]

i'm not sure but i think i found a set of 8 forms an+b which express every prime except 2,3,5.
and each of these forms have exceptions, ie values of n for which no. is composite, based on integer solutions of set of quadratic eqns in two variables. i will work them out probably in a day or two after the damned sessionals are over. is this a new approach or has someone already done this?
ps- what is sieve of what's-his-face?

 

[CRGreathouse]

i'm not sure but i think i found a set of 8 forms an+b which express every prime except 2,3,5.

By 8 you mean {1, 7, 11, 13, 17, 19, 23, 29} mod 30. Yes, all primes greater than 5 are of that form. What's more, almost all numbers of this form are composite -- only a tiny fraction are prime.

You can go a step higher if you'd like. All primes greater than 7 are {1, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 143, 149, 151, 157, 163, 167, 169, 173, 179, 181, 187, 191, 193, 197, 199, 209} mod 210.

and each of these forms have exceptions, ie values of n for which no. is composite, based on integer solutions of set of quadratic eqns in two variables.

Like (a + 1)(b + 1) for positive integers a, b?

is this a new approach or has someone already done this?

It's about two to three thousand years old. I'm fairly sure it wasn't known 4000 years ago.

ps- what is sieve of what's-his-face?

The sieve of Eratosthenes.

 

[Count Iblis]

There does exist a polynomial in 26 variables that generates all the primes, http://en.wikipedia.org/wiki/Formula_for_primes#Formula_based_on_a_system_of_Diophantine_equations"

 

[chhitiz]

By 8 you mean {1, 7, 11, 13, 17, 19, 23, 29} mod 30. Yes, all primes greater than 5 are of that form. What's more, almost all numbers of this form are composite -- only a tiny fraction are prime.
well, yes. it goes something like this-
30n+7 is prime for all n except when n is of form
30k₁k₂+7k₁+k₂
30k₁k₂+17k₁+k₂+6
30k₁k₂+23k₁+29k₂+22
30k₁k₂+13k₁+19k₂+8