Wikipedia:Reference desk/Archives/Mathematics/2024 May 3

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May 3

Can Carmichael number be Lucas-Carmichael number?

Can Carmichael number be Lucas-Carmichael number?

Also, varying the signs, there are four different sequences for similar numbers:

1. squarefree composite numbers k such that p | k => p-1 | k-1
2. squarefree composite numbers k such that p | k => p-1 | k+1
3. squarefree composite numbers k such that p | k => p+1 | k-1
4. squarefree composite numbers k such that p | k => p+1 | k+1

the 1st sequence is Carmichael numbers, and the 4th sequence is Lucas-Carmichael numbers, but what are the 2nd sequence and the 3rd sequence? Are there any number in at least two of these four sequences? If so, are there any number in at least three of these four sequences? 61.224.150.139 (talk) 05:07, 3 May 2024 (UTC)

According to Lucas–Carmichael number, it is unknown whether there are any Lucas–Carmichael numbers that are also Carmichael numbers. GalacticShoe (talk) 05:52, 3 May 2024 (UTC)

2. The sequence is OEIS:A208728, and it starts $15,35,255,455,1295,2703,4355,6479,9215,\ldots$

3. The sequence is OEIS:A225711, and it starts $385,2737,6061,6721,17641,24769,25201,\ldots$

GalacticShoe (talk) 06:21, 3 May 2024 (UTC)

Do all numbers in any of these four sequences except 15 and 35 have at least three prime factors? 61.224.150.139 (talk) 06:41, 3 May 2024 (UTC)

Yes. You can show that:

1. If $p-1,q-1|pq-1$, then $p-1|q-1$ and $q-1|p-1$, implying $p-1=q-1\Rightarrow p=q$, which is disallowed.
2. If $p-1,q-1|pq+1$, then $p-1|q+1$ and $q-1|p+1$, implying either $p=q-2$ or $p=q+2$ (since they can't be equal.) The rest of this proof is left to the reader since I don't feel like writing it down, but based on the fact that $n|(n+2)^{2}\Leftrightarrow n|4$, it can be shown that $pq=15,35$ only.
3. If $p+1,q+1|pq-1$, then $p+1|q+1$ and $q+1|p+1$, implying $p+1=q+1\Rightarrow p=q$, which is disallowed.
4. If $p+1,q+1|pq+1$, then $p+1|q-1$ and $q+1|p-1$, implying $p\leq q-2$ and $p\geq q+2$, which is not possible.

GalacticShoe (talk) 07:52, 3 May 2024 (UTC)

Are all of these four sequences infinite? If so, do all of these four sequences contain infinitely many terms with exactly 3 prime factors? Also, do all of these four sequences contain infinitely many terms which are divisible by a given odd prime number? 2402:7500:943:D56F:909B:9877:85C8:AFAA (talk) 02:02, 5 May 2024 (UTC)

@GalacticShoe: 49.217.60.214 (talk) 05:02, 6 May 2024 (UTC)

It is known that there are infinitely many Carmichael numbers, and it was recently (Wright, 2016) proven that there are infinitely many Lucas–Carmichael numbers. Unfortunately, I am unsure of the other two sequences, although Wright's paper might have more information for someone more mathematically literate than I. GalacticShoe (talk) 17:14, 10 May 2024 (UTC)

If Dickson conjecture is true, then do all of these four sequences contain infinitely many terms with exactly 3 prime factors? 2402:7500:900:DEEB:B513:C07E:8EF3:8275 (talk) 04:13, 11 May 2024 (UTC)

In order for a sequence to be in both 1. and 2., this would require that all prime factors $p|k$ satisfy both $p-1|k-1,p-1|k+1\Rightarrow p-1|2\Rightarrow p=2,3$. The only squarefree composite number that is only composed of $2,3$ is $6$ which can easily be seen to not be in either sequence. Similarly, 3. and 4. would require all prime factors $p|k$ to satisfy both $p+1|k-1,p+1|k+1\Rightarrow p+1|2$ which does not hold for any primes $p$. Since 1. and 2. cannot coexist, nor can 3. and 4., this means that no number occupies three or more of the four sequences. GalacticShoe (talk) 06:25, 3 May 2024 (UTC)

Carmichael numbers are the numbers n such that $\lambda (n)$ divides n-1, where $\lambda (n)$ is the Carmichael lambda function (also called reduced totient function, since it is the reduced form of the Euler totient function) (sequence A002322 in the OEIS), and Lucas-Carmichael numbers should be the numbers n such that $\eta (n)$ divides n+1, and this $\eta (n)$ should be a reduced form of the Dedekind psi function, use the same reduce rule as the Carmichael lambda function to the Euler totient function (i.e. use the least common multiple in place of the multiplication for $\lambda (mn)$ and $\eta (mn)$ with coprime m, n), but I cannot even find this function in OEIS (it should start with (start from n=1) 1, 3, 4, 6, 6, 12, 8, 12, 12, 6, 12, 12, 14, 24, 12, 24, 18, 12, 20, 6, 8, 12, 24, 12, …) 2402:7500:943:D56F:909B:9877:85C8:AFAA (talk) 02:13, 5 May 2024 (UTC)

@GalacticShoe: 2402:7500:900:DEEB:B513:C07E:8EF3:8275 (talk) 04:09, 11 May 2024 (UTC)

Factorial & primorial on wikipedia

I know the factorial notation n!. Recently I cam across 5# which I was unfamiliar with. Not knowing its name, (primorial), it proved hard to track down. I searched in wikipedia and Google for "n#" which seemed like the best bet. Both converted it to "n" and reported stuff about the 14th letter of the alphabet. I then thought maybe it is related to factorial, so I looked at wikipedia factorial (n! redirects to factorial on wikipedia, so that works if/when you don't know the term "factorial".)

So is there a way of making n# findable on wikipedia? If so how? -- SGBailey (talk) 21:21, 3 May 2024 (UTC)

Search for "#" and find Number sign#Mathematics. —Kusma (talk) 22:25, 3 May 2024 (UTC)

@SGBailey: "#" is WP:FORBIDDEN in page names. I'm actually impressed that a search on "#" gives Number sign as the only result. I examined redirects and saw the similar Unicode characters ﹟ and ＃. Thinking that it may help search, I have redirected N﹟ and N＃ to Primorial. The "go" feature of the search box ignores "#" in "n#" and goes directly to N, but if you force a real search on n# then the third or fourth (it varies) result for me is now "Primorial (redirect from N﹟)". PrimeHunter (talk) 19:57, 6 May 2024 (UTC)

Excellent - Thank you. -- SGBailey (talk) 18:00, 10 May 2024 (UTC)