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Prime Numbers Beget Prime Numbers

Suppose P is a seven-digit prime number.

The consecutive digits of P can be used to form new numbers,
some of them prime → 3, 5, 13, 61, 491.

Question 1: What is the seven-digit prime P?

Question 2: Four other primes can be formed from the consecutive digits of P...what are they?

 

Source: "Intellectual Rebel" on community.livejournal.com


Hint: Make sure you understand the problem. Suppose you were given the prime number 1237. From it's consecutive digits, one can make the prime numbers 3, 7, 23, and 123 (237 is not prime). Now, back to the original problem....

 


Solution Commentary: Adaptation of HappySteve's solution (on a listserve in 2005): The 61 and a 491 imply there have to be two 1s in P. That's five digits already; the other two digits are 3 and 5, noting that the 3 must follow either 61 or 491. Thus P is a permutation of: (5)(613)(491) OR (5)(61)(4913).

The 5 cannot be the last digit since P wouldn't be prime, leaving 8 possible permutations to form P, two of which are prime:

4913561 = 17*289033
4915613 = 41*113*1061
5491361 = 19*289019
5491613
5613491 = 13*431807
5614913 = 17*330289
6135491
6154913 = 37*166349

Thus, P is either 5491613 or 6135491.

And now to list the primes found within these two primes:
5491613 → 3, 5, 13, 61, 491, 613, 1613, 9161, 54916

6135491 → 3, 5, 13, 61, 491, 613, 35491, 613549