Pythagorean Dates

This Thursday will be the last Pythagorean date we will see in this century.

The date - 5 12 13 - will be the last date this year where the day, month and year form a Pythagorean triple such that [date]² + [month]² = [year]².

(5, 12, 13) is an example of a Pythagorean triple - three positive integers a, b, and c, such that a² + b² = c². Such a triple is commonly written (a, b, c). The most famous example is (3, 4, 5).

If (a, b, c) is a Pythagorean triple, then so is (ka, kb, kc). A primitive Pythagorean triple is one in which the only positive integer that evenly divides a, b and c is 1 - the three integers in the triple must be coprime.

I got to thinking about Pythagorean triples overall, on the bus ride home, and found myself working out a few triples above and beyond the basic (3, 4, 5) relationship.

For instance, there are the triples below 100 where two numbers are adjacent - (3, 4, 5) is the only Pythagorean triple where all three numbers are adjacent:-

(03, 04, 05) --> 3² (9) + 4² (16) = 5² (25)

(05, 12, 13) --> 5² (25) + 12² (144) = 13² (169)

(07, 24, 25) --> 7² (49) + 24² (576) = 25² (625)

(08, 15, 17) --> 8² (64) + 15² (225) = 17² (289)

(09, 40, 41)

(11, 60, 61)

(13, 84, 85)

These are the other Pythagorean triples under 100:-

(12, 35, 37)

(16, 63, 65)

(20, 21, 29)

(28, 45, 53)

(48, 55, 73)

(36, 77, 85)

(33, 56, 65)

(39, 80, 89)

(65, 72, 97)

Generating Pythagorean Triples There is a formula, called Euclid's Formula, which is used to determine if three numbers (a, b, c) form a Pythagorean triple for any arbitrary integer numbers k, m and n, such that

a = k . (m² - n²)

b = k . (2mn)

c = k . (m² + n²)

There is a lot more information on Pythagorean triples here, on this Wikipedia page from which some of this information comes.