PG PRIMES

[enzocreti]

Originally Posted by lukerichards

Reducing the number of variables (but increasing the complexity), this is:

Yes these primes are not random at all but nobody is interested in it

[lukerichards]

Originally Posted by enzocreti

Yes these primes are not random at all but nobody is interested in it

By and large people will only be interested in primes of certain forms. Specifically:

• Where P+1 or P-1 is easily factored; or
• Where the form can be easily sieved using existing sieving technology

Or both of the above.

The first example lends itself to being provable with an N+1 or N-1 proof.
The second example may not be provable, but can be tested PRP relatively quickly once low-level factoring has been done.

Anything else will not really have quite the value to hunters.

[VBCurtis]

Originally Posted by enzocreti

....not random at all ....

Define these words in the mathematical context you use them in. Be precise.

You flail about declaring patterns after 5 tests, when the rest of the forum tests hundreds of thousands of items. If you think there's a pattern, define that pattern precisely, and then DO THE TESTS YOURSELF. Up to, say, 10,000. Or 50,000. See how your pattern holds. Then, let us know how well the actual data fits your definition of "not random at all".

Or, disappear with your tail between your legs when you realize patterns that appear for really small numbers aren't patterns at all. Until you fully digest how meaningless your "patterns" are, it's unlikely anyone listens to you even if you did find something of interest because so much of what you find is an artefact of only testing tiny numbers.

Perhaps a history lesson about the Mersenne primes, and how many of the first few prime powers are prime. Pattern? Or Fermat numbers, where the first 4 are prime and no others (known). Pattern?

If you only ever posted after testing 1,000 candidates for whatever "interesting" thing you found today, you and the forum would be much better off.

[enzocreti]

ok i Will keep it in mind

[enzocreti]

Originally Posted by enzocreti

ok i Will keep it in mind

Ok there are three exponents leading to a pg prime which are multiples of 215:


215, 69660, 92020


There is one 75894 leading to a pg prime which is only one far away from a multiple of 215.


Is this possible by mere chance?
Do you believe so?

[CRGreathouse]

Originally Posted by enzocreti

Ok there are three exponents leading to a pg prime which are multiples of 215:


215, 69660, 92020


There is one 75894 leading to a pg prime which is only one far away from a multiple of 215.


Is this possible by mere chance?
Do you believe so?

What's so special about 215? Was there some reason you were looking for multiples of 215, rather than multiples of 214 or 217? Because it sounds like what you're really surprised about is that three of your 37 numbers have a common factor. But I don't think that this is all that rare. If I just generate random lists of 37 distinct positive numbers up to 366770 and take the three with the largest gcd, I get (sample of 100):
113, 71, 54, 165, 66, 34, 124, 26, 64, 40, 66, 166, 45, 18, 25, 97, 197, 60, 101, 32, 52, 72, 128, 98, 48, 170, 83, 426, 25, 205, 91, 45, 49, 29, 33, 33, 22, 71, 21, 135, 89, 25, 65, 44, 64, 25, 56, 108, 101, 33, 22, 30, 30, 66, 89, 35, 71, 160, 74, 199, 26, 39, 90, 27, 27, 57, 60, 63, 42, 66, 66, 72, 46, 51, 81, 209, 34, 52, 77, 33, 33, 78, 39, 59, 370, 127, 40, 101, 40, 59, 18, 35, 131, 86, 139, 70, 129, 26, 54, 44
which doesn't make 215 seems special at all. I'm sure incorporating special divisibility facts about your numbers would make it even more typical.

PARI/GP:

gcd3max(v)=if(#v<3,return(0));if(#v==3,return(gcd(v)));my(mx=1);for(i=2,#v,for(j=1,i-1, my(g=gcd(v[i],v[j])); if(g<=mx,next); for(k=1,#v,if(gcd(g,v[k])>mx&&k!=i&&k!=j,mx=gcd(g,v[k])))));mx
randomdistinct(n,k)=if(k>n,error("need k <= n")); my(v=vector(k,i,random(n))); v=Set(v); while(#v<k, my(t=random(n)); if(!setsearch(v,t), v=setunion(v,[t])));v
vector(100,i,gcd3max(apply(n->n+1,randomdistinct(366770,37))))

[VBCurtis]

Originally Posted by enzocreti

Is this possible by mere chance?

Yes.
What other possible explanation is there? If you can't predict this *before* noticing it, then you have discovered a coincidence.

If the coincidence is so strong that you make a prediction, *and* that prediction holds for a large set of numbers, then you have perhaps made a discovery about an underlying relationship among the numbers. That's how many of us got "in to" modular arithmetic; we noticed a pattern, thought it was odd, and when looking to see if this was already known we discovered a massive branch of mathematics that nicely explained what seemed like an unlikely relationship.

In your case, as Dr Greathouse so kindly informed you, you repeatedly find minor coincidences that are entirely explained by random chance. You keep doing the factoring equivalent of rolling a die 10 times, and exclaiming that 6 came up twice in a row!!!! Not only do you exclaim you rolled 6 twice in a row, now you're trying to say that the die isn't random because you got 6 twice in a row this time, while you got 4 twice in a row last time.

These exclamations show that you lack any grasp of what "random" is. You don't even consider probability. If I roll a die 50 times and post the results of the rolls in order, I'm quite sure you can find lots of "interesting" patterns in the data. That doesn't make the data more or less random, until you can find a pattern that lets you *predict* outcomes.

[Batalov]

As a reminder - this is the second iteration of the same discussion. Deja vu?

[lukerichards]

Originally Posted by VBCurtis

Yes.
What other possible explanation is there? If you can't predict this *before* noticing it, then you have discovered a coincidence.

The other possible explanation is that there is a mathematical reasoning behind this which is beyond the scope of the OP's powers of prediction. As a teacher, I'm sure you can imagine, this happens to my pupils all the time.

For example, take a bunch of 11 year olds and ask them to make predictions about the sequence:

5n + 10

and I bet very few of them will predict that a) it will be linear and b) it will generate values in the 5 times table. This is not, of course, a coincidence.

[LaurV]

Prime numbers are not "random". They are very well defined, since thousands of years ago. The algorithm to find them all is very deterministic, and actually quite fast - thinking of a guy called Eratosthenes. Run the sieve for as long as you want, and you will find the prime you want. Of course, in all these thousands of years, we, humanity, were too stupid to find the magic formula. And probably will still be so stupid for thousands years to come. But the OP is right, there is no coincidence and no accidents in it...

Joking apart, these cranks new people coming here have no idea, and no grasp, of the magnitude of the numbers we are playing with. Talking about millions does not seems much larger than talking about hundreds... but when these are exponents, it would suffice to say that a hundred means more than all the particles in the known universe. And a hundred and one is ten times more... Grrr... I myself have no understanding, and no grasp, and my brain can not conceive such big numbers...

[lukerichards]

Originally Posted by LaurV

Talking about millions does not seems much larger than talking about hundreds... but when these are exponents, it would suffice to say that a hundred means more than all the particles in the known universe. And a hundred and one is ten times more... Grrr... I myself have no understanding, and no grasp, and my brain can not conceive such big numbers...

I mentioned a book in another thread - 'Humble Pi' by Matt Parker - I thoroughly reccommend it.

In it he says when it comes to describing the size of numbers, he talks about time.

The time now is 14:10:00 BST on 9th May 2019.

If you count 1 second from now it will still be 14:10 BST on on 9th May 2019. 1 second is the smallest 1 digit integer.

If you count up to the smallest 2-digit integer: 10 seconds, the time will still be 14:10 BST on on 9th May 2019.

If you count up to the smallest 3-digit integer: 100 seconds, the time will still be 14:11 BST on on 9th May 2019.

If you count the smallest 6-digit integer... well, that's not much bigger is it? It's only double the number of digits. 100,000 seconds from 14:10:00 on 9th May 2019 will take us to 17:46 on 10th May 2019. A whole day has passed, and then some.

Add on just 1 more digit? You'll be going until some time in the early hours of the morning of 21st May 2019. A million seconds (7 digits) takes you 11.57 days to count at a rate of 1 per second.

So a million is fine... that's under two weeks. What about a billion seconds? Only 10 digits. That will be roughly January 2051.

And what about a trillion? A number we're all familiar with when hearing about the national debt...

The smallest 13 digit number will take you until about March 31729CE. 31 thousand years from now at a rate of 1 per second. Just 13 digits. Thirteen digits.