What is the least prime which has 32 1-bits?

Knowing $2^{32}-1$ isn't prime, I found $2^{33}-1 - 2^{23}=8581545983 $ just by starting with a string of 33 ones and replacing a one with a zero starting from second most significant bit and moving right.

Robert Israel's OEIS link (https://oeis.org/A061712) doesn't list any non-trivial math properties, but this seems pretty trivial to write a program for. Start with a long string of binary ones ($2^k-1$) and try setting 1 one to zero, 2 ones to zero with $2^{k+1}-1$, etc.


Since you know about Mersenne primes, you know that $2^{32} - 1$ is composite, right? It's not a safe assumption in the age of Betsy DeVos.

So the next best thing would be a string of thirty-two $1$s with a $0$ in there somewhere. Obviously $2^{33} - 2$ is even, but take the binary representation $111111111111111111111111111111110$ and rotate carry right within the $33$-bit word, $111111111111111111111111111111101$, $111111111111111111111111111111011$, etc., until you find a prime.

Easy as pie, right?

Tags:

Binary

Entropy

Prime Numbers

Bit Strings

Related

What are all the unramified extensions of $\mathbb{Q}_p$? Verify vs Prove Evaluate $\int_0^1 \frac{\mathrm{d}x}{\sqrt{1-x^2}}\frac{x }{1-k^2x^2}\log\left(\frac{1-x}{1+x}\right)$ Why isn't the area of a square always greater than the length of one of its sides? Does zero curl imply a conservative field? Prove that $f(ax + (1-a)y) = \frac{1}{y-x}\int_x^y f(t)dt$ implies $a = \frac{1}{2}$ Determining locus of a complex number Prove that a degree-$6$ polynomial has exactly $2$ real roots Integration of $\int_{-1}^{1}\frac{dx}{(x-a)\sqrt{1-x^2}}$ using Residue Calculus Prove that primes of the form $2n^2+10n+15$ are also of the form $36k+7$ If $\sin^{-1} (x) + \sin^{-1} (y)+ \sin^{-1} (z)=\dfrac {\pi}{2}$, prove that $x^2+y^2+z^2+2xyz=1$ How to prove this definite integral does not depend on the parameter?

Recent Posts