Prime (4436714570233827*2^2552+1)

[enzocreti]

4436714570233827*2^52+1 is prime
4436714570233827*2^252+1 is prime
4436714570233827*2^2552+1 is prime


Is it possible with Pfgw to find if it exists another prime of the form


4436714570233827*2^255...552+1?

[enzocreti]

(4436714570233827*2^2552+1)=p is prime, a Proth prime.


The number of decimal digits of p is 28^2=784


p is congruent to (28+1=29) mod 28.


Nice property?

[LaurV]

yes

[enzocreti]

this is a Proth prime with a special form...
but there are arbitrary primes p such that p-1 is congruent to sqrt(number of decimal digits of p) mod sqrt(number of decimal digits of p)?

[CRGreathouse]

Originally Posted by enzocreti

this is a Proth prime with a special form...
but there are arbitrary primes p such that p-1 is congruent to sqrt(number of decimal digits of p) mod sqrt(number of decimal digits of p)?

Let's rephrase. Are there infinitely many integers d such that there is a prime \(p\equiv 1 \pmod d\) in the range
$$
10^{d^2-1} < p < 10^{d^2}?
$$

Seems certain, I'll leave it to others to decide if it is provable.

[VBCurtis]

I don't understand how you can post thread after thread of the same sort of "wow, I found this, can you find more?" without learning how to use the tools that find more. Please, learn pfgw. Maybe learn newpgen to create files of candidates while sieving out ones with small factors.

I hope you've at least concluded that you can find a prime for lots and lots of combinations fixing three of {a,b,c,d} in a*b^c+d (subject to d not divisible by b) by iterating over the letter you don't fix. There are exceptions, and they're somewhat rare; you keep exhibiting items that aren't rare, with marvel that they exist. Why shouldn't they exist?

[enzocreti]

Originally Posted by CRGreathouse

Let's rephrase. Are there infinitely many integers d such that there is a prime \(p\equiv 1 \pmod d\) in the range
$$
10^{d^2-1} < p < 10^{d^2}?
$$

Seems certain, I'll leave it to others to decide if it is provable.

And is it possible to find in a smart way another proth-prime with this property?

[VBCurtis]

Yes.

[Batalov]

I am tired of moving a dozen threads from enzocreti from the wrong place to the right place - which is his blog area.

I will simply remove next ones that will be posted int he wrong place. Fair warning.

In addition: enzocreti - why do you start a new thread for every amazing new thingy? Just start one thread - "My amazing findings" and post all of them there. They are all the same - based on the strong law of small numbers. You can find a thousand more of these every day. We don't need a thousand threads "full of sound and fury, signifying nothing." You are very high maintenance.

[Uncwilly]

Originally Posted by Batalov

I am tired of moving a dozen threads from enzocreti from the wrong place to the right place - which is his blog area.
....

You are very high maintenance.

Maybe Enzo needs a vacation.....

[Batalov]

A single thread would be just fine.

Shirley, it will not be a newspaper column of Martin Gardner quality, but it will be in the same vein. He definitely enjoys it, and that's a great thing.