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Why 2^p-1 and not 2^p-2?

Why is 2[SUP]p[/SUP]-1 special, and not 2[SUP]p[/SUP]-2?

 ET_ 2019-01-06 19:43

[QUOTE=Blackadder;505144]Why is 2[SUP]p[/SUP]-1 special, and not 2[SUP]p[/SUP]-2?[/QUOTE]

Because 2[sup]p[/sup]-2 is even?

 Batalov 2019-01-06 19:45

In what sense special?
2[SUP]p[/SUP]-2 is special in its own way, too

 science_man_88 2019-01-06 19:59

[QUOTE=Batalov;505146]In what sense special?
2[SUP]p[/SUP]-2 is special in its own way, too[/QUOTE]

it's twice a repunit, it shows an iteration formula for Mersenne numbers, it's 6 mod 8 for p>2 . etc.

[QUOTE=ET_;505145]Because 2[sup]p[/sup]-2 is even?[/QUOTE]
Is that rhetorical? I really should get back into my math books.

[QUOTE=Batalov;505146]In what sense special?
2[SUP]p[/SUP]-2 is special in its own way, too[/QUOTE]
Special as in: why are we calculating for 2[sup]p[/sup]-1 and not 2[sup]p[/sup]-2 as well. I guess the latter can't be prime?

 science_man_88 2019-01-06 20:17

[QUOTE=Blackadder;505148]Is that rhetorical? I really should get back into my math books.

Special as in: why are we calculating for 2[sup]p[/sup]-1 and not 2[sup]p[/sup]-2 as well. I guess the latter can't be prime?[/QUOTE]

no number divisible by a number other than 1 or itself is prime 2[SUP]p[/SUP]-2 = 2(2[SUP]p-1[/SUP]-1) therefore is divisible by 2 except when p-1 = 1 and therefore p=2 -- 2[SUP]p-1[/SUP]-1 = 1

[QUOTE=science_man_88;505151]no number divisible by a number other than 1 or itself is prime. Now, 2[SUP]p[/SUP]-2 = 2(2[SUP]p-1[/SUP]-1) therefore is divisible by 2

there is only one exception: when p-1 = 1 (and p=2) -- 2[SUP]p-1[/SUP]-1 = 1 doesn't hurt primality.[/QUOTE]

That makes sense. Thank you.

 Batalov 2019-01-06 20:44

[QUOTE=ET_;505145]Because 2[sup]p[/sup]-2 is even?[/QUOTE]
Just to state the obvious, what Luigi 'ET_' said:
the people-speak word 'even' means (in math-speak) 'divisible by 2'. No more, no less.

If something is even - it cannot be prime, except if something is '2' itself.

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