Legend of Lucas and Lehmer

«Lucas»

From primality tests of the two math experts, you may read in reverse to know Number Theory.

In the progress of history, to find a series of Mersenne primes, it will be involed in a sequence with infinite numbers linked in line,

and further, recovering a correlation from the sequence's numbers, and then, matching up all Mersenne's from the relation formula.

Number Theory thus will show on the stage, in order to looking for useful materials from the complicated pile of numbers.

Lucas thus from Fibonacci sequence derives a formula to enable including multi-sequences, but it still can not match the prime's positions.

Lehmer

Lehmer then from Lucas sequence adjusts the formula's parameters, and also proves the new sequence has met the requirement.

Further, if you are familiar with these parameters, that means you have realized this Number Theory.


From Number Theory itslef, it reflects people view on numbers; more impurities at origin, more complication afterward.

Our site submits A New Thinking of Lucas-Lehmer Sequence; when the accumulation of knowledge makes more overwhelmed, then next, we might think about how to see much clearly.

Mersenne Prime

«Fermat»

Filled all bits by value 1, and undivided by any other integer, the number is a Mersenne prime.

Why is that named Mersenne? the erudite monk, only with pens and papers repeatedly counting off and checking on, felt in this kind of happiness.

Now, by fast computing of computers, everyone is busy breaking the world record for the largest known primes.

How to find? Lucas-Lehmer simple sequence is the current adoption; however, to see through it, you have to know many mathematicians.

Euler

Our site provides one new equation with simple proving; as long as heard about two ancestors, Fermat and Euler, you may easily understand hereof.

If b_c= b₁ (mod M_c), where b_k+1= (b_k)² -2, b₀= -(1/3 +3) (mod M_c), then M_c is a prime.