Problems & Puzzles: Conjectures

Conjecture 29. The Frank Buss's  B(n) function

Frank Buss has defined a recursive function B(n) that - conjecturally - generates:

• only primes

• none of these twice

• all the primes

This is the Buss's recursive function:

• f(1)=1

• B(n)= nxtprime{f(n)+1)} - f(n)

• f(n)=f(n-1).B(n-1)

The first few members of the sequence are:

My first comment to him was:

Question: Can you disapprove the Buss's Conjecture? that is to say:

a) find a composite B(n)
b) find one pair (n, m) such that B(n)=B(m) if n & m are distinct?
c) find a prime non-listed in the sequence B(n)
 

________
 * See (EIS A067836,  A062894 )

...It is fairly easy to see that any two members of the Buss sequence B(n) are coprime. Suppose that p is a prime factor of B(n) (I do not assume that B(n) is always prime) so that p also divides f(m) for all m > n. Consider any value of m > n, then p divides q - f(m) where q is the next prime larger than f(m)+1. But p | f(m) so q > f(m)+1 > p. Hence we cannot have p | q and so we cannot have p | B(m) = q - f(m). This implies in particular that the B(n) are all distinct...

"...about the conjecture 29 I have tested it up to n=603. No composite number has been found. ( See attached file for the values of B(n)). The first primes that have not been generated up to n=603 are: 503, 677, 811, 911, 997, 1163, 1193, 1303, 1367, 1459, 1493, 1567 ....... Only for curiosity. The function B(n) tend in average to follow the curve: 2.1*n^1.22, ..."

So, the first follow-up question for this puzzle is find the n value such that B(n)=503

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Robert Smith wrote (3/6/6):

I thought I would try to find 503 as part of this conjecture. I have taken n now on to 666, and of the listed gaps only 1163 is eliminated at n=610.

Regarding the conjecture that the series only includes primes, I could add the following:

Primorials are associated with large prime gaps, as no number in the range p#+2 to p#+p can be prime.

Let the largest value of B(n) to date be x. In the first 650n x=28559

f(n) is a deficient form of the primorial x#, being the product of distinct primes, but not necessarily all of them up to x.

Let's define the deficient factorial Q as x#/(product of the deficient primes (r[1]* r[2]*r[3]..), where the r’s are the 1st, 2nd, 3rd smallest values of B(n) which have not been generated by the Buss function to date.

Only Q + any of the r or some multiple of r’s can be prime between Q+2 and Q+x. The smallest combination of r’s is r[1]*r[2]. Up to n=603, r[1]*r[2]=503*677=340531 is significantly greater than x.=28559.

Odd integer Q+s, larger than Q+p, can be prime, as long as s is not divisible by any of the primes in the B(n) series to date. The next prime after Q+2 will be defined by s= a prime not in the series B(n) to date, or by a multiple of the r’s . Again, the smallest non-prime s would be r[1]*r[2].

The largest prime smaller than 340531 is 340519, which is the 29223rd prime. Only 603 primes are in B(n) series, therefore all of the 28620 values of Q+(one of the primes not in the B(n) series) must be composite, and Q+340531 must also be prime.

The chances of a non prime value is therefore negligible and will only become non negligible if either these low values of r[1] and r[2] persist for many, many iterations. However, negligible is not the same as zero. A string of 29223 composites might just be round the corner!

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On December 15, 2014, Dana Jacobsen wrote:

I have calculated the first 3334 terms of OEIS A067836  (the B(n) of conjecture 29.  B(3289) = 503, B(3048) = 1193, B(2887) = 2027.  No composites (and no duplicates) to this point.  The largest value to this point is B(3249) = 145069.  The smallest values not present: 3659, 4243, 5581, 6217, 6323, 6529, 6833, 6961, 7129, 7219, 7243.
 

f(3289) is 13361 digits, and it takes about 1 minute to find B(3289) from f(3289) since this is a very small gap.  I used the Perl ntheory module for the calculations.  The next_prime function uses an ES BPSW test, but it wouldn't be hard to run more tests on the values.
 

I suppose I should update the OEIS b-files.
 

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