Say when you get a "x", let "k" is the first occurance in which
(7x+k-1) and (k) is NOT in lowest terms
we can say 7x+k-1 = r*p and k=r*q in which r is a prime number
then you will get 7x+r*q-1=r*p and r=(7x-1)/(p-q)
now you get 7x-1 has a factor of r
In your question, x must meet the requirement:
The smallest prime factor of 7x-1 should be bigger than 301. Of course x should be at least 43
and you will easily exclude some cases like x=2n+1,3n+1,5n+3 beause in these cases 7x-1 contains small prime factor 2,3,5
Then the only work you should do is to check
44,50,54,56,60 and you will find 7*54-1=377=13*29 and 7*56-1=391=17*23 which should also be excluded
The remaining numbers are the answer
7*44-1 = 307; 7*50-1 = 349; 7*60-1=419. They are all prime numbers
(7x+k-1) and (k) is NOT in lowest terms
we can say 7x+k-1 = r*p and k=r*q in which r is a prime number
then you will get 7x+r*q-1=r*p and r=(7x-1)/(p-q)
now you get 7x-1 has a factor of r
In your question, x must meet the requirement:
The smallest prime factor of 7x-1 should be bigger than 301. Of course x should be at least 43
and you will easily exclude some cases like x=2n+1,3n+1,5n+3 beause in these cases 7x-1 contains small prime factor 2,3,5
Then the only work you should do is to check
44,50,54,56,60 and you will find 7*54-1=377=13*29 and 7*56-1=391=17*23 which should also be excluded
The remaining numbers are the answer
7*44-1 = 307; 7*50-1 = 349; 7*60-1=419. They are all prime numbers