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The greatest number theorist in mathematical universe, Leonhard Euler had discovered some formulas and relations in number theory, which were based on practices and were correct to limited extent. The prime generating equation by Euler is a binomial which is actually very specific and yields more primes than any other relations out there in number theory. Euler told that the equation yields many prime numbers with the values of x being input from x=0 to x=k-2; k being a prime.
Let’s see how many primes we can get by using different values of k and x:
|Serial Number||Value of k (prime)||Value of x (from x=0 to x=k-2)||Value of f(x)=(x^2+x)+k||Not a Prime?|
The above table yields many prime numbers, which again can be put at the place of k and so on the table can be progressed.
According to Euler, 41 was the most appropriate value of k yielding more prime numbers than any other k. In the list below, each value of f(x) is a prime for k=41:
So, the Euler’s Prime Generating Equation can be written as
; where x is an integer ranging from 0 to 39.
Wait. What if we increase the value of x beyond the limit of 39? What will we get?
The next values of f(x) in this series would be 1681, 1763, 1847, 1933, 2021, 2111, 2203, 2297, 2393, … .
Are all these prime numbers too? The answer is no. 1681 is not a prime number, neither are 1763 and 2021. Though all others are prime numbers.
Fermat Number, a class of numbers, is an integer of the form .
For example: Putting in we get , , , etc.
Fermat observed that all the integers were prime numbers and announced that is a prime for each natural value of .
In writing to Prof. Mersenne, Fermat confidently announced:
I have found that numbers of the form are always prime numbers and have long since signified to analysts the truth of this theorem.
However, he also accepted that he was unable to prove it theoretically. Euler in 1732 negated Fermat’s fact and told that are primes but is not a prime since it is divisible by 641.
Euler also stated that all Fermat numbers are not necessarily primes and the Fermat number which is a prime, might be called a Fermat Prime. Euler used division to prove the fact that is not a prime. The elementary proof of Euler’s negation is due to G. Bennett.
|The Fermat number is divisible by i.e., .|
Factorising in such a way that
Subtracting from 641, we get .
Now again, equation (1) could be written as
Mathematics is on its progression and well developed now but it is yet not confirmed that whether there are infinitely many Fermat primes or, for that matter, whether there is at least one Fermat prime beyond . The best guess is that all Fermat numbers are composite (non-prime).
A useful property of Fermat numbers is that they are relatively prime to each other; i.e., for Fermat numbers , .
Following two theorems are very useful in determining the primality of Fermat numbers:
|For , the Fermat number is prime|
Euler- Lucas Theorem
|Any prime divisor of , where , is of form .|
Fermat numbers () with are prime; with have completely been factored; with have two or more prime factors known; with have only one prime factor known; with have no factors known but proved composites. has not yet been proved either prime or composite.
- Video: Documentary on Proof of Fermat’s Last Theorem (wpgaurav.wordpress.com)