A property about the order of divisors of (Mq-1)/2
 

Let q prime and [tex]\large M_q=2^q-1[/tex] prime.

Let define: [tex]\large order(d,2)[/tex] the least i such that [tex]\large 2^i \equiv \pm 1 \ \pmod{d}[/tex].

[tex]\large \varphi(d)[/tex] is the Euler function (number of numbers lower than d and coprime with d).

Then, if [tex]\ \large d \ \mid \ (M_q-1)/2[/tex] with d>1 , then [tex]\large order(d,2) \ \mid \ q-1[/tex] and [tex]\large \ order(d,2) \ \mid \ \varphi(d)[/tex] .


Is that well-known ?

(It is a consequence of a paper I'm reading now)


Examples:
[tex]\large q=5 \ , \ (M_q)/2=3*5[/tex]
[tex]\large d=3 : \ order(d,2)=1 ,\ \varphi(d)=2[/tex]
[tex]\large d=5 : \ order(d,2)=2 ,\ \varphi(d)=4[/tex]
[tex]\large d=15 : \ order(d,2)=4 ,\ \varphi(d)=4[/tex]

[tex]\large q=7 \ , \ (M_q)/2=3^2*7[/tex]
[tex]\large d=3 : \ order(d,2)=1 ,\ \varphi(d)=2[/tex]
[tex]\large d=7 : \ order(d,2)=3 ,\ \varphi(d)=6[/tex]
[tex]\large d=9 : \ order(d,2)=6 ,\ \varphi(d)=6[/tex]
[tex]\large d=21 : \ order(d,2)=6 ,\ \varphi(d)=12[/tex]
[tex]\large d=63 : \ order(d,2)=6 ,\ \varphi(d)=36[/tex]

[tex]\large q=13 \ , \ (M_q)/2=3^2*5*7*13[/tex]
[tex]\large d=13 : \ order(d,2)=6 ,\ \varphi(d)=12[/tex]
etc
[tex]\large d=65 : \ order(d,2)=6 ,\ \varphi(d)=48[/tex]
[tex]\large d=91 :\ order(d,2)=12 ,\ \varphi(d)=72[/tex]
etc
[tex]\large d=4095 : \ order(d,2)=12 ,\ \varphi(d)=1728[/tex]

Tony

A property of divisors of Mq ?
 

I've checked only with some q primes (109, 103, 101, 97, 83, 37, 13, 7) , but it seems that:

[tex]\large \ d \ \mid \ M_q \ \Rightarrow \ order(d,2) = q \ [/tex] , where [tex]\large \ order(d,2)[/tex] is the least i such that [tex]\large \ 2^i \ \equiv 1 \ \pmod{d}[/tex].

Is this property already well-known ?

Tony

[QUOTE=T.Rex]I've checked only with some q primes (109, 103, 101, 97, 83, 37, 13, 7) , but it seems that ... [/QUOTE] This is stupid. It is the definition of d being a divisor of Mq. (I should learn more elementary Number Theory, spend more time searching by my-self, and thus say less garbage on this forum. Isn't it, Bob ?)

What about the [B]property in first post of this thread[/B] ?
(This property is not mine. So it is probably less stupid than my own tries.)

(In the examples, [tex]\large (M_q)/2[/tex] should be: [tex]\large (M_q-1)/2[/tex].)

Tony

[QUOTE=T.Rex]Let q prime and [tex]\large M_q=2^q-1[/tex] prime.

Let define: [tex]\large order(d,2)[/tex] the least i such that [tex]\large 2^i \equiv \pm 1 \ \pmod{d}[/tex].

[tex]\large \varphi(d)[/tex] is the Euler function (number of numbers lower than d and coprime with d).

Then, if [tex]\ \large d \ \mid \ (M_q-1)/2[/tex] with d>1 , then [tex]\large order(d,2) \ \mid \ q-1[/tex] and [tex]\large \ order(d,2) \ \mid \ \varphi(d)[/tex] .

Is that well-known ?
[/QUOTE]

This looks to be simple manipulations of the definition of order. For starters, [tex] \ (M_q-1)/2 = 2^{q-1} - 1[/tex]. Then, [tex] 2^{q-1} \equiv 1 \pmod{d}[/tex] implies that the order of 2 modulo d divides q-1. The second claim is Lagrange's Theorem.

The definition of order you give is different from the commonly accepted one, which has 1 instead of [tex]\pm 1[/tex]. You may wish to consider switching.

Hope this helps,
John