Problems & Puzzles: Conjectures

Conjecture 94. Follow-up Conjecture 29

Davide Rotondo sent this follow-up to Conjecture 29

I checked the Buss conjecture but I made a variant i.e. I start from 4 and take
into consideration the next prime of the form 4k-1.
I get:
Start with f(1)=4
Calculate b(n)=nextprimeof the form 4k-1>f(n)+1 and subtract f(n)
Calculate f(n+1)=f(n)b(n)


f(n).                        minimal prime>f(n)+1.          b(n)
                              of the form 4k-1


 4,                                      7,                                 3
 12,                                   19,                                 7
 84,                                  103,                                19
 1596,                              1607,                               11
 17556,                            17579,                              23
 403788,                          403831,                             43
 17362884,                     17362987,                          103
 1788377052,                1788377083,                          31
 55439688612,              55439688679,                         67
 3714459137004,          3714459137111,                     107
 397447127659428,     397447127659579,                    151
 

Here too the primes of the form 4k-1 appear to be distinct. Are there only prime numbers of
the form 4k-1? Do all prime numbers of the form 4k-1 appear?

 

During the week from Aug, 27 to Set 3, contributions came from Emmanuel Vantieghem

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Emmanuel wrote:

Both questions have a negative answer.
 
Indeed, we have :
b(12) = 415 = 5*83.
 
The prime number 83 will never appear in the sequence
The fact that the  b(i)  are coprime to each other is proved in a similar way as in Conjecture 29 (See Felice Russo's contribution).
 

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