Is this (prime numbers) formula known ?

Godzilla · 2018-09-26, 10:33

Good morning,

Is this formula known? does it work or does it not work?

The formula is :

$((p^1*p^2)+(p^2 or p^1))-1 = prime number$

Thank you.

.

retina · 2018-09-26, 10:47

Quote:

Originally Posted by Godzilla

Good morning,

Is this formula known? does it work or does it not work?

The formula is :

$((p^1*p^2)+(p^2 or p^1))-1 = prime number$

Thank you.

.

Yes, it works perfectly.

p1=2, p2=13 ---> winner

Godzilla · 2018-09-26, 11:00

Quote:

Originally Posted by retina

Yes, it works perfectly.

p1=2, p2=13 ---> winner

and while p > 2 ?

Thanks

.

retina · 2018-09-26, 11:08

Quote:

Originally Posted by Godzilla

and while p > 2 ?

Thanks

.

You could also test it for yourself and find your own counter examples. Just a thought.

Godzilla · 2018-09-26, 11:16

Quote:

Originally Posted by retina

You could also test it for yourself and find your own counter examples. Just a thought.

You forum members do you know or not if it works?

.

Godzilla · 2018-09-26, 11:36

I found a counterexample that does not work.

CRGreathouse · 2018-09-26, 13:50

With the primes up to 10^5 this works 10240881 out of 91996872 times (about 11%).

LaurV · 2018-09-26, 13:57

Quote:

Originally Posted by Godzilla

I found a counterexample that does not work.

If you find a counterexample that works, you can get the Nobel prize...

CRGreathouse · 2018-09-26, 15:53

559526371/6161857506 for primes up to a million. (I'm done.)

Dr Sardonicus · 2018-09-26, 18:25

I'm not sure what the or means [possibilities include the bitwise OR of p1 and p2, which in Pari-GP would be bitor(p1,p2)].

For the sake of discussion I will assume it means

p1*p2 + p1 - 1 or p1*p2 + p2 - 1.

It occurred to me that one could make both expressions divisible by the same prime q. This would require that p1 == p2 (mod q). Calling the common residue class x, we have

x^2 + x - 1 == 0 (mod q).

This quadratic has two solutions (mod q) when 5 is a quadratic residue (mod q), i.e. when q == 1 or 9 (mod 10). For q = 11, the two values of x are 3 (mod 11) and 7 (mod 11).

Thus, if p1 and p2 are both congruent to 3 (mod 11) or both are congruent to 7 (mod 11), both p1*p2 + p1 - 1 and p1*p2 + p2 - 1 will be divisible by 11.

For example, we could take p1 = 3 and p2 = 47, or p1 = 7 and p2 = 29.

Dr Sardonicus · 2018-09-27, 02:23

I was thinking about the possibility that "p1 or p2" meant bitor(p1, p2), and noticed that, if 2 < p1 < p2, and 2^(r-1) < p1 < 2^r, then

bitor(p1,p2) = p2 when p2 == p1 (mod 2^r).

This would make the expression equal to

p2*(p1 + 1) - 1.

If q is a prime which does not divide p1 + 1, p1 + 1 has a multiplicative inverse (mod q), and the expression is divisible by q when p2 == (p1 + 1)^-1 (mod q).

OK, so it's easy to construct counterexamples in which

bitor(p1,p2) = p2 and q divides p1*p2 + p2 - 1,

provided q does not divide p1 + 1.

Then, I noticed something curious. The first 2 odd primes are 3 and 5. The congruence classes 3 (mod 4) and 5 (mod 8) cover the congruence classes 3, 5, and 7 (mod 8). That only leaves the class 1 (mod 8) uncovered. So it occurred to me to wonder whether it ever does get covered by classes p1 (mod 2^r).

Since the first prime congruent to 1 (mod 8) is 17 and the next is 41, things start off badly. You need to cover 8 residue classes (mod 64) and only 3 of them (17, 49, and 41) are covered. That leaves 5 classes (mod 64). Make that 10 classes (mod 128).

You have to cover 1/4 of the odd residue classes (mod 2^r) for some r (those congruent to 1 (mod 8)), and the primes congruent to 1 (mod 8) are about a quarter of the primes, and the primes keep getting thinner on the ground. So I sort of doubt the class 1 (mod 8) ever gets covered.

2018-09-26, 10:33	#1
Godzilla May 2016 163 Posts	Is this (prime numbers) formula known ? Good morning, Is this formula known? does it work or does it not work? The formula is : $((p^1*p^2)+(p^2 or p^1))-1 = prime number$ Thank you. .

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2018-09-26, 11:36	#6
Godzilla May 2016 163 Posts	I found a counterexample that does not work.

2018-09-26, 13:50	#7
CRGreathouse Aug 2006 5,987 Posts	With the primes up to 10^5 this works 10240881 out of 91996872 times (about 11%).

2018-09-26, 15:53	#9
CRGreathouse Aug 2006 5,987 Posts	559526371/6161857506 for primes up to a million. (I'm done.)

2018-09-26, 18:25	#10
Dr Sardonicus Feb 2017 Nowhere 6,221 Posts	I'm not sure what the or means [possibilities include the bitwise OR of p1 and p2, which in Pari-GP would be bitor(p1,p2)]. For the sake of discussion I will assume it means p1p2 + p1 - 1 or p1p2 + p2 - 1. It occurred to me that one could make both expressions divisible by the same prime q. This would require that p1 == p2 (mod q). Calling the common residue class x, we have x^2 + x - 1 == 0 (mod q). This quadratic has two solutions (mod q) when 5 is a quadratic residue (mod q), i.e. when q == 1 or 9 (mod 10). For q = 11, the two values of x are 3 (mod 11) and 7 (mod 11). Thus, if p1 and p2 are both congruent to 3 (mod 11) or both are congruent to 7 (mod 11), both p1p2 + p1 - 1 and p1p2 + p2 - 1 will be divisible by 11. For example, we could take p1 = 3 and p2 = 47, or p1 = 7 and p2 = 29.

2018-09-27, 02:23	#11
Dr Sardonicus Feb 2017 Nowhere 6,221 Posts	I was thinking about the possibility that "p1 or p2" meant bitor(p1, p2), and noticed that, if 2 < p1 < p2, and 2^(r-1) < p1 < 2^r, then bitor(p1,p2) = p2 when p2 == p1 (mod 2^r). This would make the expression equal to p2(p1 + 1) - 1. If q is a prime which does not divide p1 + 1, p1 + 1 has a multiplicative inverse (mod q), and the expression is divisible by q when p2 == (p1 + 1)^-1 (mod q). OK, so it's easy to construct counterexamples in which bitor(p1,p2) = p2 and q divides p1p2 + p2 - 1, provided q does not divide p1 + 1. Then, I noticed something curious. The first 2 odd primes are 3 and 5. The congruence classes 3 (mod 4) and 5 (mod 8) cover the congruence classes 3, 5, and 7 (mod 8). That only leaves the class 1 (mod 8) uncovered. So it occurred to me to wonder whether it ever does get covered by classes p1 (mod 2^r). Since the first prime congruent to 1 (mod 8) is 17 and the next is 41, things start off badly. You need to cover 8 residue classes (mod 64) and only 3 of them (17, 49, and 41) are covered. That leaves 5 classes (mod 64). Make that 10 classes (mod 128). You have to cover 1/4 of the odd residue classes (mod 2^r) for some r (those congruent to 1 (mod 8)), and the primes congruent to 1 (mod 8) are about a quarter of the primes, and the primes keep getting thinner on the ground. So I sort of doubt the class 1 (mod 8) ever gets covered.