Exponents of Known Mersenne Primes (Numerical Observations)

[kriesel]

Originally Posted by Dobri

Out of 27483 27-bit LCS-generated prime exponents,

...
only 2334 Mersenne numbers are untested and unfactored.

There's something wrong (incomplete) with your program or algorithm for determining "Unverified" in https://mersenneforum.org/showpost.p...5&postcount=42 posted 2022-10-21.
The effect is to overcount as unverified, known composite Mersenne numbers.

Perhaps you forgot about PRP verified status, such as by Double check with PRP & proof generation, which can appear on the same exponents as LL first test. The LL res64 may remain unverified while the exponent's composite status is conclusively verified by PRP3type1/proof/cert.
(PRP & proof to a reasonable power is both much more reliable, and usually ~1-4% more efficient, than following an LL first test with LL DC with LL's ~2% error rate per run.)
It's likely a similar issue appears in the 26-bit list, and maybe the rest of the bit levels posted too.
I suggest you refine your code & process and update by edit in place, to correct for PRP verified also, the existing posts' lists by bit level, and please convert from inline code segments to text file attachments, which place less of a burden on the storage space of the forum db, and scroll just fine in any decent text editor after a download. It also has the advantage of holding an entire bit level's list in a single file, not up to 3 posts.
I only spot checked the following from the 27 bit list:

74211223 Unverified has PRP/proof/cert good 2022-10-12
74211227 Unverified has PRP/proof/cert good 2022-10-13
74211721 Unverified has PRP/proof/cert good 2022-10-13
74212001 Unverified has PRP/proof/cert good 2022-10-13
74212547 Unverified has PRP/proof/cert good 2022-10-15
74213609 Unverified has PRP/proof/cert good 2022-10-19
74214377 Unverified (LLDC in progress, ok)
74216977 Unverified
74217217 Unverified
74220313 Unverified
74223341 Unverified
74223931 Unverified
74227121 Unverified
74227177 Unverified
74227987 Unverified
74228149 Unverified
74229461 Unverified
74230463 Unverified

82576807 Unverified
82577023 Unverified
82579613 Unverified
82587913 Unverified
82589917 Unverified has PRP/proof/cert good 2022-06-17

Note the certs listed above were completed days or months before the "unverified" 27 bit list was posted.

The 30-bit list looks ok; it does not contain any of the 17 exponents in https://www.mersenne.org/report_prp/...dispdate=1&B1=

[Dr Sardonicus]

I don't see the point of looking for long common substrings of bits in Mersenne prime exponents. The exponents yielding Mersenne primes are very sparse indeed. If you want a prime with a long string of bits in common with a given prime, a good place to look is at nearby primes. The reason is simple: If P is a given prime of any size, nearby primes Q generally have |Q - P| very small compared to P. That is, the number of bits in |Q - P| will be small in comparison to the number of bits in P.

So assume the number of bits in |P - Q| is very small compared to the number of bits in P. If Q > P we have the addition Q = P + (Q - P), and if Q < P we have the subtraction Q = P - (P - Q). If Q > P the only way for Q to disagree with P in bits significantly beyond those in |Q - P|, is to have a string of 1's starting in the bits of P - Q, and extending much farther into the more significant digits of P than the number of bits in Q - P. This allows a succession of carries in the addition to flip a bunch of 1's to 0's.

If Q < P what you need is for P to have a long string of zeroes starting in the bits of P - Q, and extending much farther into the significant figures of P than the number of bits in P - Q. This allows the borrows in the subtraction to flip a bunch of 0's to 1's.

So I would say it is likely that most Mersenne prime exponents have longer strings of common bits with nearby primes which are not Mersenne prime exponents, than with exponents that do yield Mersenne primes.

Similarly, if you look for long common substrings of bits in primes and the reversed string of digits of a given Mersenne prime exponent, primes near the number represented by the reversed bits should usually fill the bill. And I am guessing that these won't be Mersenne prime exponents very often.

[Dobri]

Originally Posted by kriesel

...
Perhaps you forgot about PRP verified status, such as by Double check with PRP & proof generation, which can appear on the same exponents as LL first test. The LL res64 may remain unverified while the exponent's composite status is conclusively verified by PRP3type1/proof/cert.
(PRP & proof to a reasonable power is both much more reliable, and usually ~1-4% more efficient, than following an LL first test with LL DC with LL's ~2% error rate per run.)
It's likely a similar issue appears in the 26-bit list, and maybe the rest of the bit levels posted too.
...

The issue was spotted and promptly resolved. It did not affect the subsequently posted 26-bit list.
Only 21 prime exponents of the 27-bit LCS-generated prime exponents are currently LL(or PRP without proof)-Unverified but also PRP-CERT-Verified:
M74206771, M74206889, M74207383, M74211223, M74211227, M74211721, M74212001, M74212547, M74213609, M74216977, M76795057, M82589917, M83780327, M84127793, M85912759, M86225219, M87051017, M88906319, M88911799, M100018879, and M113449253.
This number will increase as more GIMPS volunteers are starting to test the LL-Unverified prime exponents with PRP-CERT instead of LL-DC.

[Dobri]

Originally Posted by Dr Sardonicus

...
So I would say it is likely that most Mersenne prime exponents have longer strings of common bits with nearby primes which are not Mersenne prime exponents, than with exponents that do yield Mersenne primes.
...

I agree with the quoted statement. However, the intended direction of study is based on the assumption that the LCS binary strings of the set of prime exponents of known Mersenne primes are self-sufficient to construct the binary string of the prime exponent of the next Mersenne prime (if any) without borrowing binary strings outside from said set.

[Dobri]

A bit of English grammar, it should be "outside of" (or "aside from") instead of "outside from" in the previous post #59.

[Dobri]

Here is a list of the base-10 palindromic prime exponents which were LCS-generated in previous posts:

M102484201 (Factored),
M113565311 (Unverified),
M122232221 (Verified),
M137282731 (Untested),
M149535941 (Factored),
M186565681 (Factored),
M192191291 (Factored),
M195151591 (Factored),
M332000233 (Untested),
M335181533 (Untested),
M335333533 (Factored),
M363010363 (Factored),
M702010207 (Factored),
M905232509 (Factored), and
M908616809 (Untested).

[Dobri]

Let's represent the 51 prime exponents of known Mersenne primes as 27-bit binary strings:

{{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,1},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,1},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,0,1},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,1,0,0,1},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,1,0,1,1},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,1,1},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,1,1,1,1,1,1,1},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,1,1,0,1,1},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,1,1,0,1,0,0,1},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,1,0,0,1,0,0,0,1},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,1,1,1,0,1},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,1,0,0,0,1,1,1},{0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,0,1,1,0,0,1},{0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,1,0,1,1,0,1,0,1,0,1},{0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,1,1,1,1,0,0,1,1,0,1},{0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,1,0,1,1,1,1,0,0,0,0,1},{0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,1,0,0,1,1,0,0,0,1,0,1},{0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0,1},{0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,1,1,0,1,1,1,0,1,0,0,0,1},{0,0,0,0,0,0,0,0,0,0,1,0,1,0,1,0,0,0,0,1,1,1,0,0,0,1,1},{0,0,0,0,0,0,0,0,0,0,1,1,0,1,0,1,1,1,1,1,0,1,0,0,1,1,1},{0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,1,1,1,0,1,0,0,0,1},{0,0,0,0,0,0,0,0,0,1,1,0,1,0,0,1,1,0,0,0,0,0,1,1,0,1,1},{0,0,0,0,0,0,0,1,0,1,1,1,0,0,0,1,1,0,0,0,1,1,0,0,1,1,1},{0,0,0,0,0,0,0,1,1,0,1,0,0,0,1,1,1,0,1,0,0,1,0,1,0,0,1},{0,0,0,0,0,0,1,0,0,1,1,0,0,1,1,0,0,0,1,0,0,1,1,1,0,1,1},{0,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,1,1,1,0,1},{0,0,0,0,0,1,0,1,1,0,1,0,1,1,0,1,0,0,1,1,1,0,1,1,1,0,1},{0,0,0,0,0,1,0,1,1,1,0,0,0,0,1,1,0,1,0,0,1,0,0,0,0,0,1},{0,0,0,0,1,1,0,1,0,1,0,0,1,1,0,0,1,0,0,1,0,1,1,0,0,0,1},{0,0,0,1,1,0,0,1,1,0,1,0,1,1,1,1,1,0,1,0,0,1,0,0,1,0,1},{0,0,1,0,1,0,0,0,0,0,0,0,1,0,1,1,1,1,1,1,0,1,0,1,0,1,1},{0,0,1,0,1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,1,1,1,0,0,1,1,1},{0,0,1,1,0,0,0,1,1,0,0,0,0,1,1,0,0,0,1,1,0,0,1,0,1,1,1},{0,0,1,1,1,0,0,1,1,1,1,1,1,1,0,0,1,1,1,1,0,0,1,1,0,0,1},{0,0,1,1,1,1,1,0,0,0,1,0,0,1,0,1,1,0,0,0,0,0,0,0,0,0,1},{0,1,0,0,0,1,1,0,1,1,0,1,1,1,1,0,1,1,1,0,0,1,1,1,0,1,1},{0,1,0,1,0,0,0,1,0,1,0,1,0,1,1,0,0,0,1,0,1,0,1,1,0,0,1},{0,1,0,1,0,0,1,0,0,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0,0,0,1},{0,1,1,0,1,1,1,0,0,1,1,0,1,0,0,0,0,0,1,1,1,1,0,1,0,0,1},{1,0,0,0,1,1,0,1,1,0,0,0,1,0,1,0,0,0,0,0,0,1,1,0,0,0,1},{1,0,0,1,0,0,1,1,0,1,0,0,1,1,1,1,0,1,1,0,0,0,1,0,1,0,1},{1,0,0,1,1,1,0,1,1,0,0,0,0,1,1,1,0,0,0,1,1,1,0,1,1,0,1}}

Then Gram-Schmidt orthogonalization can be performed:

{{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1/Sqrt[2],0,0,1/Sqrt[2],0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,Sqrt[2/3],-(1/Sqrt[6]),0,0,1/Sqrt[6],0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1/Sqrt[3],1/Sqrt[3],0,0,-(1/Sqrt[3]),0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}}

(* Wolfram code *)
MpData = {2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 42643801, 43112609, 57885161, 74207281, 77232917, 82589933};
nMp = Length[MpData]; base = 2; intlenmax = Length[IntegerDigits[MpData[[nMp]], base]]; MpDataBinary = ConstantArray[0, {nMp, intlenmax}];
ic = 0; While[ic < nMp, ic++; MpDataBinary[[ic]] = IntegerDigits[MpData[[ic]], base, intlenmax];];
Print[MpDataBinary]; Print[Orthogonalize[MpDataBinary]];

After removing the all-zero rows, the following 27 rows remain after said orthogonalization:

{{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0},
{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1},
{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0},
{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0},
{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0},
{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0},
{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0},
{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0},
{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2^-1/2,0,0, 2^-1/2,0,0,0,0,0,0,0},
{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,(2/3)^1/2,-6^-1/2,0,0, 6^-1/2,0,0,0,0,0,0,0},
{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3^-1/2,3^-1/2,0,0,-3^-1/2,0,0,0,0,0,0,0},
{0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0},
{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0},
{0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0},
{0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0},
{0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},
{0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},
{0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},
{0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},
{0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},
{0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},
{0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},
{0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},
{0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},
{0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},
{0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},
{1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}}.

[Dobri]

Obviously, the determinant of the 3×3 matrix {{0, 1/Sqrt[2], 1/Sqrt[2]}, {Sqrt[2/3], -(1/Sqrt[6]), 1/Sqrt[6]}, {1/Sqrt[3], 1/Sqrt[3], -1/Sqrt[3]}} is equal to 1.

See some of the properties of said matrix such as characteristic polynomial, eigenvalues, etc., at https://www.wolframalpha.com/input?i...t%5B3%5D%7D%7D.

[Dobri]

The presence of {0,2^-1/2,0,0, 2^-1/2} is due to M1279,
the presence of {(2/3)^1/2,-6^-1/2,0,0, 6^-1/2} is due to M2203, and
the presence of {3^-1/2,3^-1/2,0,0,-3^-1/2} is due to M3217,

[Dobri]

List of x-coordinates of the prime exponents of known Mersenne primes in Ulam clockwise square spiral (see also OEIS A174344):

{1,0,-1,0,1,-2,-2,2,-1,5,-5,6,4,-5,-1,18,-1,-3,33,-32,-49,9,-30,71,74,-76,82,46,-166,182,99,-374,464,536,-591,863,-869,-967,148,-2291,-374,1717,-1018,-2854,1501,-3265,-3283,-3804,-4307,-4394,-2733}

List of y-coordinates of the prime exponents of known Mersenne primes in Ulam clockwise square spiral (see also OEIS A268038):

{-1,-1,0,1,-2,-1,1,-3,-4,-4,2,-1,11,12,-18,23,-24,28,4,33,36,-50,-53,14,-19,29,105,-147,113,-99,232,-435,359,-561,554,266,-136,1320,-1835,-1004,2451,-2548,-2757,-1461,-3048,-364,-3030,-307,1978,-421,-4544}

List of {x, y}-coordinates of the prime exponents of known Mersenne primes in Ulam clockwise square spiral:

{{1,-1},{0,-1},{-1,0},{0,1},{1,-2},{-2,-1},{-2,1},{2,-3},{-1,-4},{5,-4},{-5,2},{6,-1},{4,11},{-5,12},{-1,-18},{18,23},{-1,-24},{-3,28},{33,4},{-32,33},{-49,36},{9,-50},{-30,-53},{71,14},{74,-19},{-76,29},{82,105},{46,-147},{-166,113},{182,-99},{99,232},{-374,-435},{464,359},{536,-561},{-591,554},{863,266},{-869,-136},{-967,1320},{148,-1835},{-2291,-1004},{-374,2451},{1717,-2548},{-1018,-2757},{-2854,-1461},{1501,-3048},{-3265,-364},{-3283,-3030},{-3804,-307},{-4307,1978},{-4394,-421},{-2733,-4544}}

(* Wolfram code *)
SetDirectory[NotebookDirectory[]]; fname1 = NotebookDirectory[] <> "MersenneUlamX&Y.jpg"; fname2 = NotebookDirectory[] <> "MersenneUlamXY.jpg";
MpData = {2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 42643801, 43112609, 57885161, 74207281, 77232917, 82589933};
nMp = Length[MpData]; MpSx = ConstantArray[0, nMp]; MpSy = ConstantArray[0, nMp]; MpSxy = ConstantArray[0, {nMp, 2}];
count = 1; n = 0; xc = 0; yc = 0; While[n < MpData[[nMp]], n++; ac = Mod[Floor[Sqrt[4*(n - 1) + 1]], 4]*Pi/2; xc = xc + Sin[ac]; yc = yc + Cos[ac]; If[n == MpData[[count]], MpSx[[count]] = xc; MpSy[[count]] = yc; count++;];];
ic = 0; While[ic < nMp, ic++; MpSxy[[ic, 1]] = MpSx[[ic]]; MpSxy[[ic, 2]] = MpSy[[ic]];];
Print[MpSx]; Print[MpSy]; Print[MpSxy];
Show[ListLinePlot[{MpSx, MpSy}, PlotRange -> All, Frame -> True, PlotLegends -> {"x", "y"}]]
Export[fname1, Show[ListLinePlot[{MpSx, MpSy}, PlotRange -> All, Frame -> True, PlotLegends -> {"x", "y"}]]]
Show[ListLinePlot[MpSxy, PlotRange -> All, Frame -> True]]
Export[fname2, Show[ListLinePlot[MpSxy, PlotRange -> All, Frame -> True]]]

[Dobri]

The attached graph shows the sorted list of Log[Abs[x]] and Log[Abs[y]] values of x- and y-coordinates of the prime exponents of known Mersenne primes in Ulam clockwise square spiral.

SetDirectory[NotebookDirectory[]];
fname = NotebookDirectory[] <> "MersenneUlamX&YSortLogAbs.jpg";
MpSx = {1, 0, -1, 0, 1, -2, -2, 2, -1, 5, -5, 6, 4, -5, -1, 18, -1, -3, 33, -32, -49, 9, -30, 71, 74, -76, 82, 46, -166, 182, 99, -374, 464, 536, -591, 863, -869, -967, 148, -2291, -374, 1717, -1018, -2854, 1501, -3265, -3283, -3804, -4307, -4394, -2733};
MpSy = {-1, -1, 0, 1, -2, -1, 1, -3, -4, -4, 2, -1, 11, 12, -18, 23, -24, 28, 4, 33, 36, -50, -53, 14, -19, 29, 105, -147, 113, -99, 232, -435, 359, -561, 554, 266, -136, 1320, -1835, -1004, 2451, -2548, -2757, -1461, -3048, -364, -3030, -307, 1978, -421, -4544};
nMp = Length[MpSx]; ic = 0; While[ic < nMp, ic++; MpSx[[ic]] = Log[Abs[MpSx[[ic]]]]; MpSy[[ic]] = Log[Abs[MpSy[[ic]]]];];
MpSx = Sort[MpSx]; MpSy = Sort[MpSy];
Show[ListLinePlot[{MpSx, MpSy}, PlotRange -> All, Frame -> True, PlotLegends -> {"x", "y"}]]
Export[fname, Show[ListLinePlot[{MpSx, MpSy}, PlotRange -> All, Frame -> True, PlotLegends -> {"x", "y"}]]]