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## Main Question or Discussion Point

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.?

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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.?

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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)

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CRGreathouse

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As many as you'd like. 2n + 1, for example.is there any other general form of prime no.s known except 6n+/-5 and 6n+/-1.

n = 141.is there any general form of n such that 6n+/-5 or 6n+/-1 is a composite no.?

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.

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but, then, if i am not wrong, this series doesn't have a pattern, does it?n in {1, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 19, 20, 21, 22, 23, 24, 25, ...}

.

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.

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CRGreathouse

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Wow, impredicativity in real life!but, then, if i am not wrong, this series doesn't have a pattern, does it?

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

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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.

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CRGreathouse

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

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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.

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CRGreathouse

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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.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.

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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

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CRGreathouse

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

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

I intended the second, but both work.and, is it r^(2+pi/2) or r^2+pi/2

- #12

matt grime

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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.

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

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.

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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?

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?

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CRGreathouse

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

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.

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

It's about two to three thousand years old. I'm fairly sure it wasn't known 4000 years ago.is this a new approach or has someone already done this?

The sieve of Eratosthenes.ps- what is sieve of what's-his-face?

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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"

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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

30k_{1}k_{2}+7k_{1}+k_{2}

30k_{1}k_{2}+17k_{1}+k_{2}+6

30k_{1}k_{2}+23k_{1}+29k_{2}+22

30k_{1}k_{2}+13k_{1}+19k_{2}+8

well, yes. it goes something like this-

30n+7 is prime for all n except when n is of form

30k

30k

30k

30k

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