Talk:Coprimality

(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 $\scriptstyle n\,$ th term is the product of the first $\scriptstyle n\,$ primes (Cf. primorial), plus 1. Again, some terms are composite. Another example involving multiplication is that of the Kummer numbers (A057588) for which the $\scriptstyle n\,$ th term is the product of the first $\scriptstyle n\,$ primes (Cf. primorial), minus 1.

Just two more examples: $\scriptstyle n!+1\,$ (A038507) and $\scriptstyle n!-1\,$ (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)