General form of prime no.s

In summary: 30k1k2+11k1+7k2+10 30k1k2+29k1+11k2+6 30k1k2+1k1+23k2+26 30k1k2+19k1+13k2+14 30n+11 is prime for all n except when n is of form 30k1k2+11k1+k2+5 30n+13 is prime for all n except when n is of form 30k1k2+19k1+k2+14 30n+17 is prime for all n except
  • #1
chhitiz
221
0
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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  • #2
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)
 
  • #3
chhitiz said:
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.

chhitiz said:
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.
 
  • #4
CRGreathouse said:
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.
 
  • #5
chhitiz said:
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.
 
  • #6
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.
 
  • #7
chhitiz said:
i have no idea what that line between n and k stands for.

"such that"
 
  • #8
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.
 
  • #9
chhitiz said:
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.
 
  • #10
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
 
  • #11
chhitiz said:
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.

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

I intended the second, but both work.
 
  • #12
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.
 
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  • #13
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?
 
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  • #14
chhitiz said:
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.

chhitiz said:
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?

chhitiz said:
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.

chhitiz said:
ps- what is sieve of what's-his-face?

The sieve of Eratosthenes.
 
  • #16
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
 
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1. What is the general form of a prime number?

The general form of a prime number is 6n + 1 or 6n - 1, where n is a positive integer. This means that all prime numbers can be expressed as either one more or one less than a multiple of 6.

2. How do you determine if a number is prime using its general form?

To determine if a number is prime using its general form, you can plug the number into the formula 6n + 1 or 6n - 1. If the resulting number is divisible by any number other than 1 and itself, then the original number is not prime. If the resulting number is not divisible by any number other than 1 and itself, then the original number is prime.

3. Are all prime numbers in the general form of 6n + 1 or 6n - 1?

No, not all prime numbers can be expressed in the general form of 6n + 1 or 6n - 1. There are some exceptions, such as 2 and 3, which are prime but do not fit this form.

4. Can the general form of prime numbers be used to find all prime numbers?

No, the general form of prime numbers cannot be used to find all prime numbers. While it can help identify some prime numbers, there are many other prime numbers that do not fit this form and would be missed.

5. Is there a specific pattern or rule for the general form of prime numbers?

There is no specific pattern or rule for the general form of prime numbers. It is simply a useful formula that can help identify some prime numbers, but it does not apply to all prime numbers.

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