Primes of the form 4*10^n+1 and primes of the form 16*100^n+1

enzocreti · 2019-05-05, 09:09

Here the problem:

With Pari/Gp I found that 4*10^n+1 is prime for the following values of n (serach limit n=4005$):

$0,1,2,3,13,229,242,309,957,1473,1494,3182,3727$

On the other hand, I found that, squaring 4 and 10, the number 16*100^n+1 is prime with the same search limit $n=4005$ for the following values of $n$:

$0,1,2,9,18,29,34,39,42,47,75,89,95,343,406,420,1154,1265,1442,3067,4002$

I would have expected more primes of the shape 4*10^n+1 than primes of the form 16*100^n+1, because of the growth-rate. Instead it is the contrary. Is there any mathematical reason for this phenomenon and do you think that in the long run, the lead switches?

(There is a typo in the title...16*100^n+1 is correct)

axn · 2019-05-05, 09:51

FWIW, 056806 has more terms for the first sequence.

enzocreti · 2019-05-05, 10:03

Ok but why 16*100^n+1 seems to give more primes than 4*10^n+1?
With Pfgw we could go further in the investigation...maybe there is a reason due to the factors.

axn · 2019-05-05, 10:28

Quote:

Originally Posted by enzocreti

Ok but why 16*100^n+1 seems to give more primes than 4*10^n+1?
With Pfgw we could go further in the investigation...maybe there is a reason due to the factors.

I don't know. Perhaps random luck caused the second sequence to have lot of small primes. Perhaps there may be some differences due to the factors, as you suggested. I would suggest to try to search the second sequence a bit higher (say upto 30,000) and see if it continues to find more primes.

enzocreti · 2019-05-05, 10:53

the factors of 16*100^n+1 cannot be of the form 4k+3?

enzocreti · 2019-05-05, 10:56

for 7785 pfgw found another prp!!!

enzocreti · 2019-05-05, 12:34

in the case of numbers 16*100^n+1 should be many restrictions not present in the case 4*10^n+1 so far example they cannot be divisible by 7 or by primes 8k+1

Dr Sardonicus · 2019-05-05, 13:19

Quote:

Originally Posted by enzocreti

the factors of 16*100^n+1 cannot be of the form 4k+3?

This is certainly part of the answer -- along with the fact that numbers 4*10^n + 1 can have factors of this form (e.g. p = 7, 19, 23, 47, 59).

I don't know how relevant this is, but when n = 4k, 4*10^n + 1 has the Aurifeuillian (algebraic) factorization

4*10^4k + 1 = L*M =

(2*10^2k - 2*10^k + 1)*(2*10^2k + 2*10^k + 1).

2019-05-05, 09:09	#1
enzocreti Mar 2018 2·5·53 Posts	*Primes of the form 410^n+1 and primes of the form 16100^n+1* Here the problem: With Pari/Gp I found that 410^n+1 is prime for the following values of n (serach limit n=4005$): $0,1,2,3,13,229,242,309,957,1473,1494,3182,3727$ On the other hand, I found that, squaring 4 and 10, the number 16100^n+1 is prime with the same search limit $n=4005$ for the following values of $n$: $0,1,2,9,18,29,34,39,42,47,75,89,95,343,406,420,1154,1265,1442,3067,4002$ I would have expected more primes of the shape 410^n+1 than primes of the form 16100^n+1, because of the growth-rate. Instead it is the contrary. Is there any mathematical reason for this phenomenon and do you think that in the long run, the lead switches? (There is a typo in the title...16100^n+1 is correct) Last fiddled with by enzocreti on 2019-05-05 at 09:24*

2019-05-05, 10:03	#3
enzocreti Mar 2018 2×5×53 Posts	OK Ok but why 16100^n+1 seems to give more primes than 410^n+1? With Pfgw we could go further in the investigation...maybe there is a reason due to the factors.

2019-05-05, 10:53	#5
enzocreti Mar 2018 2·5·53 Posts	reason the factors of 16*100^n+1 cannot be of the form 4k+3?

2019-05-05, 10:56	#6
enzocreti Mar 2018 1000010010₂ Posts	7785 another prp found for 7785 pfgw found another prp!!!

2019-05-05, 12:34	#7
enzocreti Mar 2018 1000010010₂ Posts	explanation in the case of numbers 16100^n+1 should be many restrictions not present in the case 410^n+1 so far example they cannot be divisible by 7 or by primes 8k+1

2019-05-05, 09:51	#2
axn Jun 2003 2²×3×421 Posts	FWIW, 056806 has more terms for the first sequence.

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