Mathematics: Interesting facts and numbers

Some beautifull examples with multiplication
;
12345679 x 9 = 111111111;
12345679 x 8 = 98765432;

--------------------------------------------------------------------------------
An interesting number 2519
Let us look at 2519 Mod n (n = 2, ... 10).
2519 Mod n means reminder portion of (2519\n), where "\" is the integer division.

2519 Mod 2 = 1; 2519 Mod 3 = 2; 2519 Mod 4 = 3; 2519 Mod 5 = 4; 2519 Mod 6 = 5; 2519 Mod 7 = 6; 2519 Mod 8 = 7; 2519 Mod 9 = 8; 2519 Mod 10 = 9.

2519 = 1259 x 2 + 1; 2519 = 839 x 3 + 2; 2519 = 629 x 4 + 3; 2519 = 503 x 5 + 4; 2519 = 419 x 6 + 5; 2519 = 359 x 7 + 6; 2519 = 314 x 8 + 7; 2519 = 279 x 9 + 8; 2519 = 251 x 10 + 9.

-------------------------------------------------------------------------------
Example of wrong proof
Find a mistake in the following chain of arguments, pretending to prove that 2=1

1)	Let a = b
2)	Multiply 1) by a a2 = ab
3)	Add a2 – 2ab to both parts of 2) a2 + a2 – 2ab = ab + a2 – 2ab
4)	3) could be simplified: 2a2 – 2ab = a2 – ab
5)	It is the same as 2(a2 – ab) = 1(a2 – ab)
6)	Reduce 5) by (a2 – ab). 2=1

Where is a mistake?
Mistake is in the 6^th step.
We can not divide by (a² – ab) because a²– ab = 0.
a = b, so a²– ab = 0.

---------------------------------------------------------------------------------
An interesting fact about primes
Mathematicians of XVIII^th century proved that numbers 31; 331; 3331; 33331; 333331; 3333331; 33333331 are primes. It was a big temptation to think that all numbers of such kind are primes. But the next number is not a prime.
333333331 = 17 * 19607843

--------------------------------------------------------------------------------
An elegant proof that
It is obvious that 1 = (2 -1).
= * (2 -1) = (1 + 2 + 2² + ... + 2ⁿ) * (2 -1) =
(2 + 2² + 2³ ... + 2ⁿ⁺¹) - (1 + 2 + 2² + ... + 2ⁿ) = 2ⁿ⁺¹ - 1.