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

Question

On the many prime number investigation sites across the web I haven't been able to find the answer. Also my math isn't good enough to compute it from first principles.

So, what is the least prime that has 32 1-bits? Of course this refers to its base 2, i.e., binary representation. Programmer-speak for this would be, "32 'set' bits.'

Layperson explanation of the logic behind finding the answer would be an appreciated bonus.

[edit:] Summary of Answers

The answer is $8581545983$. By assuming the answer has just a single 0-bit, we discover that there are $5$ such ($33$-bit) primes with exactly $32$ 1-bits. Failing that, we would have gone on to try two 0-bits, etc.

The subtle point that confused both of the experts (and myself) is that even though we're seeking the lowest value here, it's incorrect to scan the substitution positions in right←to←left order of increasing significance.

The mis-intuition is caused by the fact that each trial replaces a $1$ with a $0$ at the given bit position—and not the other way around—so the higher the bit significance, the greater the reduction of numeric power. If we were to begin evaluating at the LSB (from the bottom of the following list) and proceed leftwards (upwards), we'd be fruitlessly starting at the candidate with the highest (i.e., least-reduced) overall value.

So the correct scan of values instead proceeds by testing 33 0-bit positions from left→to→right, as follows (from top-to-bottom):

011111111111111111111111111111111  4294967295   numeric value
101111111111111111111111111111111  6442450943     ↓   ↓   ↓
110111111111111111111111111111111  7516192767
111011111111111111111111111111111  8053063679
111101111111111111111111111111111  8321499135
111110111111111111111111111111111  8455716863
111111011111111111111111111111111  8522825727
111111101111111111111111111111111  8556380159
111111110111111111111111111111111  8573157375
111111111011111111111111111111111  8581545983  prime       ←- answer
111111111101111111111111111111111  8585740287
111111111110111111111111111111111  8587837439  prime
111111111111011111111111111111111  8588886015
111111111111101111111111111111111  8589410303  prime
111111111111110111111111111111111  8589672447
111111111111111011111111111111111  8589803519
111111111111111101111111111111111  8589869055
111111111111111110111111111111111  8589901823  prime
111111111111111111011111111111111  8589918207
111111111111111111101111111111111  8589926399
111111111111111111110111111111111  8589930495
111111111111111111111011111111111  8589932543
111111111111111111111101111111111  8589933567
111111111111111111111110111111111  8589934079
111111111111111111111111011111111  8589934335
111111111111111111111111101111111  8589934463
111111111111111111111111110111111  8589934527
111111111111111111111111111011111  8589934559
111111111111111111111111111101111  8589934575
111111111111111111111111111110111  8589934583  prime       ←- wrong answer
111111111111111111111111111111011  8589934587
111111111111111111111111111111101  8589934589     ↑   ↑   ↑
111111111111111111111111111111110  8589934590  0-bit significance

I would be surprised if the logic behind finding the answer is not "brute force", maybe with a little bit of number theory thrown in to exclude cases before you check them. — Arthur, Feb 05 '18 at 07:35
@GlennSlayden Robert is #3 in rep overall on the site, so probably not — qwr, Feb 05 '18 at 07:40
An answer consisting only of a link is generally frowned upon. — Robert Israel, Feb 05 '18 at 07:42
@qwr Plus, the table he linked only goes up to 31. I had asked for 32. — Glenn Slayden, Feb 05 '18 at 07:42
@GlennSlayden it links to https://oeis.org/A061712/b061712.txt which is probably more than you'll ever need — qwr, Feb 05 '18 at 07:43
The b-file goes up to $3320$. The Maple program returns $8581545983$ in a fraction of a second. — Robert Israel, Feb 05 '18 at 07:44
@RobertIsrael Ok, thanks; I didn't know that site was so clever. — Glenn Slayden, Feb 05 '18 at 07:46

qwr · Accepted Answer · 2018-02-05T08:13:27.463

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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.

edited Feb 05 '18 at 08:13

answered Feb 05 '18 at 07:39

qwr

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But you should have found $8581545983 = 2^{33} -2^{23}-1$. You searched in the wrong direction. – Robert Israel Feb 05 '18 at 07:47
@RobertIsrael you're right. I fixed in my post. – qwr Feb 05 '18 at 07:53

score 1 · Answer 2 · answered Feb 05 '18 at 22:39

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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?

answered Feb 05 '18 at 22:39

The Short One

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This is the same mistake I made, you need to start on left and go right – qwr Feb 06 '18 at 00:59
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@TheShortOne Sorry for being dense (in my case, despite DeVos, I fear): what makes you sure that the desired prime is guaranteed to have exactly one zero. I see that https://projecteuclid.org/download/pdf_1/euclid.em/999188636 proves that "there is no prime number with precisely 2^m bits, exactly two of which are zero bits," but what if none of the 33 candidates your method examines happen to be prime? Does the cited proof (or perhaps something else) require the target prime to have exactly one zero? (Obviously, it does: 1_1111_1111_1111_1111_1111_1111_1111_0111) – Glenn Slayden Feb 06 '18 at 02:32
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@Glenn Then you try two zero bits, three zero bits, etc. Mwahahahahahahaha! – The Short One Feb 08 '18 at 22:40
2^32-1 = 3 * 01010101010101010101010101010101. – gnasher729 Aug 28 '21 at 23:33

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

2 Answers2