primes of the form 2^a*3^b*5^c*p+9001

[enzocreti]

129001 is an example of a prime of the form
2^a*3^b*5^c*p+9001 with a, b and c integers and p prime.
these primes divide numbers of the form 2^m-1



3671951225346187430519619001 is another example which divides 2^69660-1

can you find other primes of this form?

129001 divides 2^m-1 for which integer values of m?

[Dr Sardonicus]

Originally Posted by enzocreti

129001 is an example of a prime of the form
2^a*3^b*5^c*p+9001 with a, b and c integers and p prime.

129001 - 9001 = 120000.

120000 = 2^4 * 3^1 * 5^4.

What is p?

[enzocreti]

[QUOTE=Dr Sardonicus;531518]129001 - 9001 = 120000.

120000 = 2^4 * 3^1 * 5^4.

What is p?[/QUOTE


129001 divides the numbers of the form 2^(860k)-1


p is a prime or 1

[Dylan14]

Originally Posted by Dr Sardonicus

129001 - 9001 = 120000.

120000 = 2^4 * 3^1 * 5^4.

What is p?

Not quite. 2^4*3^1*5^4 = 30000.

2^4*3^1*5^4*4+9001 is 129001, but p = 4 = 2^2, so this wouldn't fit the form. However, we can rewrite this:

2^6*3^1*5^4*1+9001 is 129001, and p = 1, which is allowed.

[Dylan14]

Originally Posted by enzocreti

can you find other primes of this form?

I used the following ABC file:

ABC2 2^6*3^1*5^4*$a+9001
a: primes from 0 to 1000000

and I got 11421 primes using this file and the following command line:

pfgw64 -f10 enzocreti.txt

This took less than 10 minutes to do. I am not posting the file of the primes here, it is 277 KB, and I don't want to waste bandwidth here.

[Dr Sardonicus]

I found all primes of the form N + 9001 and N*p + 9001 up to the limit 1000000, where N is divisible by 30, and the only primes dividing N are 2, 3, and 5; and p is a prime greater than 5. There are 4652 such primes. The smallest is 9091 and the largest is 999961. I'm not going to post the 60K text file.

[LaurV]

Another time when a mad man throws a stone in the lake and 10 wise men work hard to take it out... This guy is trolling, and the best way is to leave him alone.

Any odd number n, prime or not, divides some 2^m-1, in that case the smallest such m is called the "order of n (mod 2)", and all the others are multiples of it. Picking some n with a special form from all odd natural numbers is just a "random activity". You can use a pair of dices for it... They are infinite in number...