What I refer to as the *p+1 problem* is this conjecture:

Conjecture:Every rationalzcan be represented in the formz=(p+1)/(q+1)withpandqbeing prime integers.

I have verified this conjecture for *integers* to 10^{9}.
Some calculations and conjectures related to this computation are at
the
Journal of Integer Sequences, Vol. 4.

I have also verified this conjecture for all rationals *z*
with 0 ≤ z ≤ 1 with denominator not more than 1000.

Let *f(a,b)* = the minimal positive integer *m*
such that *am-1* and *bm-1* are both prime.
Then the conjecture implies that *f(a,b)* is well-defined for all rationals
*a/b* in simplest form.

Here is a plot of *f(a,b)* for 1 ≤ *b* &le 1000,
1 ≤ *a* ≤ *b*, with *(a,b)=1*.
The horizontal axis is *b* and the vertical is *f(a,b)*
(so there are multiple points for each *b*).
The cyan points are *f(a,b)* while the yellow points
are the mean values of *f(a,b)* for each *b*.

Next is a plot of just the averages.
We can see there is a general increasing trend.
I expect *f(a,b)* = O(log^{2} *b*), at least on average,
but, of course, it has not been proved that *f(a,b)* is even finite.

Since *bm-1* is more likely to be prime
if *b* has a lot of small prime factors, we would
expect that *f(a,b)* would be smaller, on average,
for those *b* with lots of small prime factors.
Here is a plot of the average values of *f(a,b)*
(as in the plot above)
versus *sigma(b)/b*, a measure of the "abundance"
of *b*. It does seem to indicate that
large values of *sigma(b)/b* tends to keep
*f(a,b)* small.

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