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Notice to Mariners (22 ways to be).

notice to mariners.

A square of an integer can never end with a digit that is 2,3,7 or 8.
can only end with a digit that is 0,1,4,5,6,9.
other hand, considering the last two digits can end only in these ways:

00
01, 21, 41, 61, 81
04, 24, 44, 64, 84 25

16, 36, 56, 76, 96
09, 29, 49, 69, 89.

So if you see a whole number that ends in one of these 22 ways, is not a true square, beware!

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footsteps of Fermat (anything from trinkets ........)

The theory numbers "are the integers 1,2,3 ,........ So here we will treat integers.
odd numbers divided by 4 to give the rest a rest or damage 3.
The primes> 2 are all odd, therefore, divided by 4 to give a rest (in this case are said to be of the form 4k + 1) rest or give 3 (are of the form 4k + 3).
Pierre De Fermat was the first to say that every prime of the form 4k + 1 can be written uniquely as the sum of two squares.
Examples: 17 = 1 ² + 4 ²
13 = 2 ² + 3 ²
29 = 2 ² + 5 ²
41 = 4 ² + 5 ²

(Instead, the numbers of the form 4k + 3, first or not first, you do not can never write as the sum of two squares, and this can be demonstrated in a simple manner).
Fermat This finding is fascinating because it is known that prime numbers can not be broken down into factors, but he has shown us the way to decompose a single prime number, which is not the factorization but it is the decomposition in the sum of two squares. What to say instead of the primes of the form 4k + 3?
For the moment I found a way to break down, similar to that shown by Fermat, not a prime number of the form 4k + 3, but the form 8k + 3.
and it is this: each prime type 8k + 3 can be written uniquely as the sum of a
squared plus twice a square (n = a ² + 2b ² ).
Esempi: 11 = 3 ² + 2 ·1 ²
19 = 1 ² + 2 · 3 ²
43 = 5 ² + 2 ·3 ²
59 = 3 ² + 2 ·5 ²

Per il momento custodisco la dimostrazione gelosamente per me (nessuno così si potrà impadronire in modo facile indebitamente del mio lavoro), ma vi invito a trovare un controesempio (può sempre darsi che io abbia sbagliato qualcosa senza accorgermene infatti, nessuno è perfetto), cioè un numero primo any of the form 8k + 3, which is not expressible uniquely as the sum of a squared plus twice a square.