Cullen number

In mathematics, a Cullen number is a natural number of the form ${\displaystyle n\cdot 2^{n}+1}$ (written ${\displaystyle C_{n}}$). Cullen numbers were first studied by Fr. James Cullen in 1905. Cullen numbers are special cases of Proth numbers.

In 1976 Christopher Hooley showed that the natural density of positive integers ${\displaystyle n\leq x}$ for which C_n is a prime is of the order o(x) for ${\displaystyle x\to \infty }$. In that sense, almost all Cullen numbers are composite. Hooley's proof was reworked by Hiromi Suyama to show that it works for any sequence of numbers n · 2^n+a + b where a and b are integers, and in particular also for Woodall numbers. The only known Cullen primes are those for n equal:

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