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  "body": "\n## Statement\n\nA conjecture related to the generalized repunit primes: (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 $b$)\n\nFor any integer $b$, which satisfies the conditions:\n\n1. $$|b|\u003e1.$$\n1. $b$ is not a perfect power. (since when $b$ is a perfect $r$th power, it can be shown that there is at most one $n$ value such that $\\frac{b^n-1}{b-1}$ is prime, and this $n$ value is $r$ itself or a [[root|TTEA3KA5KJ]] of $r$)\n1. $b$ is not in the form $-4k^4$. (if so, then the number has aurifeuillean factorization)\n\nhas generalized repunit primes of the form\n\n$$R_p(b)=\\frac{b^p-1}{b-1}$$\n\nfor prime $p$, the prime numbers will be distributed near the best fit line\n\n$$Y=G \\cdot \\log_{|b|}\\left( \\log_{|b|}\\left( R_{(b)}(n) \\right) \\right)+C,$$\n\nwhere limit $n\\rightarrow\\infty$, $G=\\frac{1}{e^\\gamma}=0.561459483566...$\n\nand there are about\n\n$$\\left( \\log_e(N)+m \\cdot \\log_e(2) \\cdot \\log_e \\big( \\log_e(N) \\big) +\\frac{1}{\\sqrt N}-\\delta \\right) \\cdot \\frac{e^\\gamma}{\\log_e(|b|)}$$\n\nbase-*b* repunit primes less than *N*.\n\n- $e$ is the [[base of natural logarithm|0MVCFMW7S4]].\n- $\\gamma$ is Euler–Mascheroni constant.\n- $\\log_{|b|}$ is the logarithm in base $|b|$\n- $R_{(b)}(n)$ is the $n$th generalized repunit prime in base*b* (with prime *p*)\n- $C$ is a data fit constant which varies with $b$.\n- $\\delta=1$ if $b\u003e0$, $\\delta=1.6$ if $b\u003c0$.\n- $m$ is the largest natural number such that $-b$ is a $2^{m-1}$th power.\n\nWe also have the following 3 properties:\n\n1. The number of prime numbers of the form $\\frac{b^n-1}{b-1}$ (with prime $p$) less than or equal to $n$ is about $e^\\gamma \\cdot \\log_{|b|}\\big(\\log_{|b|}(n)\\big)$.\n1. The expected number of prime numbers of the form $\\frac{b^n-1}{b-1}$ with prime $p$ between $n$ and $|b| \\cdot n$ is about $e^\\gamma$.\n1. The probability that number of the form $\\frac{b^n-1}{b-1}$ is prime (for prime $p$) is about $\\frac{e^\\gamma}{p \\cdot \\log_e(|b|)}$.\n",
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        "locator": "Repunit, section Repunit primes \u003e The generalized repunit conjecture",
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