Maths and Computing:

Square numbers

A number is a perfect square if it is the result of multiplying an integer (n) by itself. (n × n or n²)

List of square numbers

Sum of odd numbers

n² is the sum of the first n odd numbers.

Square number properties

Last digit

The last digit of a square number must be 0, 1, 4, 6 or 9

Difference of two squares

a² - b² = (a - b)(a + b)

Sum of the first n square numbers

Sum of the first n odd numbers

The product of four consquetive positive integers is square

n(n+1)(n+2)(n+3) + 1 = (n² + 3n + 1)²

Pythagoras's Theorem

For any right-angled triangle, The square of the length of the hypotenuse is equal to the sum of the squares of the other two sides.

Pythagorean triplets

• If two of the sides of a triangle have integer values, and using Pythagoras's theorem, the resulting third side is also have an integer value, then the lengths of the three sides form a Pythagorean triplet
• The triplet consist of three positive integers a, b and c such that a² + b² = c²

List of Pythagorean triplets: a² + b² = c²

This tool finds all of the differnet Pythagorean triplets for the value of c between a given range of values

List of Pythagorean triplets: (a,b,c)

Triplets where c is prime

Triplets where a is odd and a = n, b = (n² - 1)/2 and c = (n² +1)/2,

(3,4,5)	32 + 42 = 52
(5,12,13)	52 + 122 = 132
(7,24,25)	72 + 242 = 252
(9,40,41)	92 + 402 = 412
(11,60,61)	112 + 602 = 612
(13,84,85)	132 + 842 = 852

Triplets where a is even and a = 2n, b = n²+1 and c = n²+1 < < < < < < < <

(4,3,5)	42 + 32 = 52
(6,8,10)	62 + 82 = 102
(8,15,17)	82 + 152 = 172
(10,24,26)	102 + 242 = 262
(12,35,37)	122 + 352 = 372
(14,48,50)	142 + 482 = 502
(16,63,65)	162 + 632 = 652
(18,80,82)	182 + 802 = 822

No paired triplets or triplets with a GCF greater than 1