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Talk:Coprimality

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(Not so...) pairwise coprime infinite sequences

I removed the following text (which I previously added, wrongfully) since these sequences definitions DO NOT guarantee that they are pairwise coprime infinite sequences (although it seems that their terms are pairwise coprime with a high probability). They are NOT pairwise coprime infinite sequences!

The Euclid numbers (A006862) form yet another example, since the  th term is the product of the first primes (Cf. primorial), plus 1. Again, some terms are composite. Another example involving multiplication is that of the Kummer numbers (A057588) for which the  th term is the product of the first primes (Cf. primorial), minus 1.

Just two more examples: (A038507) and (A033312).

Daniel Forgues 19:14, 23 April 2012 (UTC)

Haha! Check

Hisanori Mishima, PI Pn + 1 (n = 1 to 110)

for the prime factorization of the Euclid numbers (A006862), a(7) is not coprime with a(17)

a(7) = 510511 = 19 * 97 * 277
a(17) = 1922760350154212639071 = 277 * 3467 * 105229 * 19026377261

Daniel Forgues 19:14, 23 April 2012 (UTC)

Check

Hisanori Mishima, PI Pn - 1 (n = 1 to 110)

for the prime factorization of the Kummer numbers (A057588), a(35) is not coprime with a(44)

a(35) = 1492182350939279320058875736615841068547583863326864530409 = 673 * 448045542064369 * 4948626474214096948642213863754187837657
a(44) = 198962376391690981640415251545285153602734402721821058212203976095413910572269 = 673 * 65473937 * 566471804985844321 * 7970932666248247010325264452352519508898124959389

Daniel Forgues 19:22, 23 April 2012 (UTC)

Check

Hisanori Mishima, n! + 1 (n = 1 to 100)

for the prime factorization of the n! + 1 numbers (A038507), a(16) is not coprime with a(18)

a(16) = 20922789888001 = 17 * 61 * 137 * 139 * 1059511
a(18) = 6402373705728001 = 19 * 23 * 29 * 61 * 67 * 123610951

and check

Hisanori Mishima, n! - 1 (n = 1 to 100)

for the prime factorization of the n! - 1 numbers (A033312), a(15) is not coprime with a(29)

a(15) = 1307674367999 = 17 * 31 * 31 * 53 * 1510259
a(29) = 8841761993739701954543615999999 = 31 * 59 * 311 * 26156201 * 594278556271609021

The same could be said for (Hisanori Mishima, WIFC (World Integer Factorization Center))

A049650 Compositorial + 1 (a(10) = 17281= 11 * 1571; a(12) = 207361= 7 * 11 * 2693)
A060880 Compositorial − 1 (a(14) = 2903039= 17 * 170767; a(22) = 115880067071999= 17 * 6816474533647)

and

A?????? Compositorial + Next Composite (obviously, pairwise noncoprime with very high probability!)
A?????? Compositorial − Next Composite (obviously, pairwise noncoprime with very high probability!)

and so on...

But I didn't succeed in finding noncoprime pairs for those two sequences

A060881 Primorial + Next Prime (pairwise coprime with very high probability!)
A060882 Primorial − Next Prime (pairwise coprime with very high probability!)

by looking at

Hisanori Mishima, PI Pn + NextPrime (n = 1 to 100)
Hisanori Mishima, PI Pn - NextPrime (n = 1 to 100)

although I suspect that the strong law of small numbers might apply here...

Daniel Forgues 19:40, 23 April 2012 (UTC)