Biography

(1870

1871)

√ 5 f n = ( ( 1 + √ 5 ) / 2 ) n − ( ( 1 − √ 5 ) / 2 ) n √5 f_{n} = ((1 + √5)/2)^{n} - ((1 - √5)/2)^{n} √ 5 f n ​ = ( ( 1 + √ 5 ) / 2 ) n − ( ( 1 − √ 5 ) / 2 ) n .

1876

2 127 − 1 2^{127} - 1 2 1 2 7 − 1

1930

S 2 = 4 , S 3 = 14 , S 4 = 194 , . . . S_{2} = 4, S_{3} = 14, S_{4} = 194, . . . S 2 ​ = 4 , S 3 ​ = 1 4 , S 4 ​ = 1 9 4 , . . .

n > 2 , S n n > 2, S_{n} n > 2 , S n ​

S n = S n − 1 2 − 2 S_{n} = S_{n-1}^{2} - 2 S n ​ = S n − 1 2 ​ − 2 .

M p = 2 p − 1 M_{p} = 2^{p} - 1 M p ​ = 2 p − 1

p > 2 p > 2 p > 2

M p M_{p} M p ​

S p S_{p} S p ​

S 127 S_{127} S 1 2 7 ​

M 127 M_{127} M 1 2 7 ​

M 127 M_{127} M 1 2 7 ​

M 127 M_{127} M 1 2 7 ​

S 127 S_{127} S 1 2 7 ​

M 127 M_{127} M 1 2 7 ​ = 170141183460469231731687303715884105727

S 127 S_{127} S 1 2 7 ​

M 127 M_{127} M 1 2 7 ​

S 127 S_{127} S 1 2 7 ​

1883

Ⓣ ( Mathematical recreations )

(1882

94)

was educated at the École Normale in Amiens. After this he worked at the Paris Observatory under Le Verrier During the Franco-Prussian WarLucas served as an artillery officer. After the French were defeated, Lucas became professor of mathematics at the Lycée Saint Louis in Paris. He later became professor of mathematics at the Lycée Charlemagne, also in Paris.Lucas is best known for his results in number theory : in particular he studied the Fibonacci sequence and the associated Lucas sequence is named after him. He gave the well-known formula for the Fibonacci numbersLucas also devised methods of testing primality, essentially those used today. Inhe used his methods to prove that the Mersenne numberis prime . This remains the largest prime number discovered without the aid of a computer.The Lucas test for primes was refined by Lehmer in. It works as follows. Define the sequencewhere foris defined inductively byThe Lucas- Lehmer test states that a Mersenne number, with, is prime if and only ifdividesLucas showed thatis divisible bythus showing thatis prime. This was a extremely difficult calculation sinceis a big number andis unbelievably large. In factand Lucas was only able to perform the calculation since he showed thatis divisible bywithout calculatingLucas is also well known for his invention of the Tower of Hanoi puzzle and other mathematical recreations. The Tower of Hanoi puzzle appeared inunder the name of M. Claus. Notice that Claus is an anagram of Lucas! His four volume work on recreational mathematicshas become a classic.Lucas died as the result of a freak accident at a banquet when a plate was dropped and a piece flew up and cut his cheek. He died of erysipelas a few days later.