Editing Repunit (section)

=== The generalized repunit conjecture ===

A conjecture related to the generalized repunit primes:<ref>[http://primes.utm.edu/mersenne/heuristic.html Deriving the Wagstaff Mersenne Conjecture]</ref><ref>[https://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0906&L=NMBRTHRY&P=R295&1=NMBRTHRY&9=A&J=on&d=No+Match%3BMatch%3BMatches&z=4 Generalized Repunit Conjecture]</ref> (the conjecture predicts where is the next [[generalized Mersenne prime]], if the conjecture is true, then there are infinitely many repunit primes for all bases <math>b</math>)

For any integer <math>b</math>, which satisfies the conditions:

# <math>|b|>1</math>.
# <math>b</math> is not a [[perfect power]]. (since when <math>b</math> is a perfect <math>r</math>th power, it can be shown that there is at most one <math>n</math> value such that <math>\frac{b^n-1}{b-1}</math> is prime, and this <math>n</math> value is <math>r</math> itself or a [[nth root|root]] of <math>r</math>)
# <math>b</math> is not in the form <math>-4k^4</math>. (if so, then the number has [[aurifeuillean factorization]])

has generalized repunit primes of the form

:<math>R_p(b)=\frac{b^p-1}{b-1}</math>

for prime <math>p</math>, the prime numbers will be distributed near the best fit line

:<math>
Y=G \cdot \log_{|b|}\left( \log_{|b|}\left( R_{(b)}(n) \right) \right)+C,
</math>

where limit <math>n\rightarrow\infty</math>, <math>G=\frac{1}{e^\gamma}=0.561459483566...</math>

and there are about

:<math>
\left( \log_e(N)+m \cdot \log_e(2) \cdot \log_e \big( \log_e(N) \big)
+\frac{1}{\sqrt N}-\delta \right) \cdot \frac{e^\gamma}{\log_e(|b|)}
</math>

base-''b'' repunit primes less than ''N''.

*<math>e</math> is the [[e (mathematical constant)|base of natural logarithm]].
*<math>\gamma</math> is [[Euler–Mascheroni constant]].
*<math>\log_{|b|}</math> is the [[logarithm]] in [[base of a logarithm|base]] <math>|b|</math>
*<math>R_{(b)}(n)</math> is the <math>n</math>th generalized repunit prime in base''b'' (with prime ''p'')
*<math>C</math> is a data fit constant which varies with <math>b</math>.
*<math>\delta=1</math> if <math>b>0</math>, <math>\delta=1.6</math> if <math>b<0</math>.
*<math>m</math> is the largest natural number such that <math>-b</math> is a <math>2^{m-1}</math>th power.

We also have the following 3 properties:

# The number of prime numbers of the form <math>\frac{b^n-1}{b-1}</math> (with prime <math>p</math>) less than or equal to <math>n</math> is about <math>e^\gamma \cdot \log_{|b|}\big(\log_{|b|}(n)\big)</math>.
# The expected number of prime numbers of the form <math>\frac{b^n-1}{b-1}</math> with prime <math>p</math> between <math>n</math> and <math>|b| \cdot n</math> is about <math>e^\gamma</math>. 
# The probability that number of the form <math>\frac{b^n-1}{b-1}</math> is prime (for prime <math>p</math>) is about <math>\frac{e^\gamma}{p \cdot \log_e(|b|)}</math>.