Euler (1770)

Most prolific mathematician of all time, 13 children. Liked to make his own hats.

• Found a closed form for the sum of the infinite series (2) = (1/n²). This is generalized from the famous harmonic series. is of course zeta.

• Euler showed in 1735 that (2) = p²/6   (rationals <---> irrationals)

First step to showing pi to be transcendental

• Euler proved the connection of the zeta function with the series of prime numbers giving the famous relation

(s) = (1/n^s) = (1 - p^-s)^-1

Here the sum is over all natural numbers n while the product is over all prime numbers.

• Euler studied it for integral powers of s.

Tombstone inscription e^(p*i)+1 =0  (prior to a geometrical interpretation of imaginary numbers)

• (1,0 for arithmetic, p for geometry, i for algebra, e for analysis)

Aside1: Fermat conjectured 2 ^{2^n}+1 is prime (called Fermat numbers), Euler factored 2³² +1. Basis: If not prime a factor existed would have to be of the form 2⁽⁵⁺¹⁾k+1 or 64k+1

Only a few need be checked, proved not prime with 641 a Carmichael Number.
He could also have checked via division of the number of primes up to 64K,  (6,500 trial divisions, primes found by the Sieve of Eratosthenes)

Aside2:  Euler partitioned 1,000,009 into 1000² +3² and 972² +235²  and thereby also producing factors of 3413 and 293. He did this when he was in his 70s and had been blind for 10 years.

Aside3: Laplace told his students, "Liesez Euler, Liesez Euler, c'est notre maître à tous" ("Read Euler, read Euler, he is our master in everything")

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