theorem?

[xilman]

Originally Posted by LaurV

Making mistakes is not a minus, and recognizing when you made a mistake is a plus. Learning from it is a big plus.

A sign of intelligence is recognizing a mistake on subsequent occasions when you make it.

[Dr Sardonicus]

Originally Posted by wildrabbitt

Thanks.



I've been rereading Sardonicus' post again because it takes me a while to digest properly. This statement puzzles me






7 is prime 7=4*1 + 3 and q = 15 = 8*1 + 7 are not a pair of Sophie Germain Primes.

Check. q = 15 isn't prime.

Also
11 is prime 11 = 4*2 + 3 and q = 25 = 8*2 + 7 aren't a pair of Sophie Germain primes either.

Er, ah, q = 8*2 + 7 is 23, not 25. And 11 and 23 are a pair of Sophie Germain primes. And, 23 divides 2¹¹ - 1,

2¹¹ - 1 = 23*89.

[wildrabbitt]

Okay I see. Thanks for not losing your patience. For a prime p, if q = 2p + 1 is prime then, p and q are a pair of Sophie Germain primes and q always divides 2^p-1.


I was in the dark because I didn't see that you meant when q = 2p+1 is prime too and because I thought 8*2 was 18.



Well I've learnt something.


Could you possibly explain why what you said here:

If p is odd, the prime q = 2*p + 1 can only be of the form 8*n + 7.

is true.


My ignorant thinking is :


\( p=2n-1\Longrightarrow 2p+1=2(2n-1)+1=4n-1\) so I don't see why a number like 3 or 11 couldn't be a q = 2p+1.

[Uncwilly]

Originally Posted by LaurV

Making mistakes is not a minus, and recognizing when you made a mistake is a plus. Learning from it is a big plus.

Originally Posted by xilman

A sign of intelligence is recognizing a mistake on subsequent occasions when you make it.

You may not get a chance to learn from your mistakes (witness the Darwin Awards). True intelligence is learning from the mistakes of others.

[LaurV]

Originally Posted by Uncwilly

True intelligence is learning from the mistakes of others.

Where is the fun in that? haha...

Also, I am a total stupid moron by that standard... If I should have put more consideration in what other people tried to teach me along the years, since I was very young, I would be a different person today. I think that is a kind of curse of being "smart"... (are we running in circles here?), you seem to be smarter than all the other kinds in your class, you always get better marks, you always solve all the exercises faster and better, and you slowly develop this "feeling" that nobody can teach you anything which you don't already know, and you refuse to believe or take into account what other people tell you, so you slowly become that person that will not learn that the door has a frame until you bump your own head into it, regardless of how many people told you before that the door has a frame. And, as Paul said, repeatedly, if possible. Bumping the head, I mean. Before learning. I was that kind of "kid" all my life...

[LaurV]

Originally Posted by wildrabbitt

\( p=2n-1\Longrightarrow 2p+1=2(2n-1)+1=4n-1\) so I don't see why a number like 3 or 11 couldn't be a q = 2p+1.

Factors of a m=2^p-1 with an odd (not necessarily prime) exponent p must be either 1 or 7 (mod 8). (more)

[Nick]

We also turned this into a fun exercise here.

[xilman]

Originally Posted by LaurV

I think that is a kind of curse of being "smart"... (are we running in circles here?), you seem to be smarter than all the other kinds in your class, you always get better marks, you always solve all the exercises faster and better, and you slowly develop this "feeling" that nobody can teach you anything which you don't already know, and you refuse to believe or take into account what other people tell you, so you slowly become that person that will not learn that the door has a frame until you bump your own head into it, regardless of how many people told you before that the door has a frame.

That description reminds me of someone around here. Now, who can it be? Name begins with 's' I think, though I may be wrong.

[kriesel]

Originally Posted by Uncwilly

You may not get a chance to learn from your mistakes (witness the Darwin Awards). True intelligence is learning from the mistakes of others.

Or perceiving the risks beforehand and adjusting. The high energy physicists I used to encounter had a shorthand question for gauging a combination of ambition and judgment which was the sort of intelligence they were interested in for engineers working on their custom research equipment projects. Roughly, what's the riskiest thing you tried and succeeded at in your youth? One of my coworkers replied, his father owned a chemical company, and so he had access to a lot of chemicals, and made his own nitrogen triiodide for fun. (The solid form explosively decomposes if tossed onto a solid surface. Small quantities, such as would form by wetting a bit of paper with the solution it's formed in and letting it dry, used to be sold as toy explosives, about as powerful as a single capgun cap.) My answer was I was raised on a dairy farm and survived without any serious disability. (Large animals, farm equipment with missing guards or risky design, climbing to 50 feet, risk of silo gas, a relative much more interested in speed than safety, often working alone, absurdly long hours, etc. The list of close calls avoided or resolved by problem solving on the fly is considerably longer than the short injury list. It was not unusual to hear on the radio about someone nearby who gambled and lost, including teens.) Both answers seemed to satisfy them.
One of the things I like about messing with mersennes is a mistake may be fatal to the computation or proof, but not to the person, unlike endeavors like climbing Everest or running a dairy farm. (If we keep our hands and metal tools out of energized power supplies.)