Problems & Puzzles: Puzzles

Puzzle 334. Farideh & the 2004 year. Farideh Firoozbakht proposed* the following puzzle:  2004 =2^2*3*167 and the three numbers 2*2004-1,3*2004-1& 167*2004-1 are prime numbers. Q1. Find a large number m with the above property,i.e., if p is a prime divisor of m then m*p-1 is prime. (I found  a 100-digit such number.) We define a(n) as the smallest number m such that m has n  distinct prime divisors and m has the above property. The first 7 terms of {a(n)} are: 2,6,30,420,32550,410970,55137810 Q2. Can you find more terms of this sequence ? Similar property: n=295855560 ,n=2^3*3^2*5*7*11*13*821 and k*295855560+1 for k=2,3,5,7,11,13& 821 are prime numbers. Q3. Find a large number like n (number of distinct prime divisors be arbitrary). We define similarly b(n) as the smallest number m such that m has n distinct prime divisors and if p is a prime divisor of m then m*p+1 is prime. The first 7 terms of the sequence {b(n)} are: 2,6,354,210,43860,463980,189583590 Q4. Can you find more terms of this sequence? _________ *(on February, 2004, so sorry for the delay because I'm publishing this on October 2005!)

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Contributions came from Anton Vrba, J. K. Andersen, Fred Schneider and Farideh Firoozbakht:

Anton wrote:

Q1 : (smile) m=2^25964950 immediately comes to mind, which has the one prime divisor p=2 and p*m-1 has 7816230 digits and is the largest known prime number credited to Dr Martin Novak and GIMPS

 

Q2 : The 8th term could be a(8)=1109*31*17*11*7*5*3*2 = 1350063330. This is the smallest number that is square free. I have checked for non square free numbers with factors 2^i i=2..6, 3^j=2..4, 5^2 and 5^3, 7^2 and 7^3, 11^2 and their combinations leading me to believe that 1350063330 is the next term of the series

 

Q4 : The 8th term is b(8)=577*503*19*11*7*5*3*2=12738238590 such the p*m+1 is prime. Again I have checked for non square free numbers smaller than b(8).

J. K. Andersen wrote:

Q1.
With 1 prime factor: m = 2^25964950.
2*m-1 = 2^25964951-1 is the largest known prime. It has 7816230 digits.
This method works for all Mersenne primes 2^p-1 with m = 2^(p-1).

With 2 prime factors: m = 4830107*2^3321.
2*m-1 and 4830107*m-1 are both titanic primes, found and proved with PrimeForm/Gw.

Fred wrote:

For puzzle #334's question #2, I could find the minimum answer for n=8. It's 1350063330=2*3*5*7*11*17*31*1109
 

Farideh wrote:

I have found a(8), a(9), b(8) & b(9). b(8) is not 12738238590 (given by Anton).

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On March 2, 2023, Michael Branicky wrote:

I have found a(10) = 9396949341780 and a(11) = 6805591029957720, and also b(10) = 60958204112190 and b(11) = 3337686588286050

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