Fermat's Little Theorem

History: On October 18, 1640, Fermat wrote a letter to Bernhard Frenicle de Bessy (1605–
1675), an official at the French mint who was a gifted student of number theory.
In his letter, Fermat communicated the following result(given as Theorem) Fermat did not provide a proof of this result but enclosed a note
promising that he would send along a proof, provided it was not too long. This theorem is known as Fermat's theorem.

Theorem: Let p be a prime and a any integer such that p and a are coprimes.
Then a^p−1 ≡ 1 (mod p).

Information:
The first proof of Fermat's little theorem was given by Euler almost a century after Fermat's announcement. Leibniz had given same proof for Fermat's theorem almost 50 years prior to Euler but he didn't receive his share of credit.

=>This theorem can be used for questions like:
Q: Find the remainder when 24¹⁹³⁶ is divided by 17. 
Ans: Here as 24 ≡ 7 (mod 17)
Therefore 24¹⁹³⁶ ≡ 7¹⁹³⁶ (mod 17)

But by Fermat's little theorem, 7¹⁶ ≡ 1 (mod 17). 
So,=7¹⁶⁸¹²¹
7¹⁹³⁶ =7^16*121
           ≡ 1¹²¹ ≡ 1 (mod 17)

Thus, remainder is 1. 

Perfect Numbers

History:
The term was first coined by Pythagoreans. The greeks thought these numbers have mystical good powers and held them to be "good" numbers. Some biblical scholars considered 6 as a perfect number because they believed that the GOD created the world in six days and GOD's work must be perfect.

Definition:
A positive integer is a perfect number if the sum of its proper factors equals n.
First perfect number is 6 then 28 then 496........

Information:
Based on assumptions, Mathematicians of Middle Ages described:
1.There is a perfect number between any two consecutive powers of 10.
2.Perfect numbers end alternatively in 6 and 8.

Now first 6 perfect numbers are:
1.6
2.28
3.496
4.8128
5.33550336
6.8589869056

Which show both assumptions get wrong as there are no prime numbers of length of 5 digits and 5th and 6th prime numbers though end in 6 but not alternatively in 6 and 8.

For perfect numbers,
Euclid gave a formula for perfect numbers:

" If n is an integer >1 such that 2ⁿ -1 is prime, then N=2^n-1(2ⁿ-1) is a perfect number."

Which gives perfect numbers but there is a fact that no odd perfect numbers are found and there is no proof available stating that odd numbers can not be perfect.