Even numbers

The split between even and odd is the oldest classification in arithmetic, and the Greeks treated it as the first thing worth saying about a number. Euclid's Elements, compiled at Alexandria around 300 BCE, opens Book VII with a block of definitions, among them an even number as one divisible into two equal parts, and Book IX then works through a long stretch of propositions — numbers 21 through 34 — on nothing but the sums, differences, products and divisibility of even and odd quantities. Parity was not a preliminary there; it was a subject.

Behind Euclid stood the Pythagoreans. Aristotle, writing roughly a generation earlier, reports in the Metaphysics that they set odd against even in a table of ten opposed pairs, alongside limited and unlimited, one and plurality, light and darkness, male and female. Even was assigned to the unlimited side. Aristotle does not really explain why, and the usual later gloss appeals to pebble figures: odd borders laid around a growing square keep producing squares, while even ones produce an endless series of different rectangles. How much of this goes back to Pythagoras himself cannot be settled — nothing he wrote survives, and Aristotle is already describing a school rather than a man.

Around 100 CE Nicomachus of Gerasa gave the scheme its late-antique form in the Introduction to Arithmetic, subdividing even numbers into the evenly-even (the powers of two), the evenly-odd and the oddly-even. Boethius translated it into Latin, and that taxonomy travelled into the medieval quadrivium, still being taught in European universities a thousand years later. Greek writers kept this kind of work, arithmētikē, separate from logistikē, the practical business of reckoning with real quantities — which is why "arithmetic" in the ancient sense meant something closer to what we now call number theory.

Key properties

• An integer is even exactly when it is divisible by 2 — when it can be written as 2k for some integer k.
• Zero is even: 0 = 2 × 0, and it sits between the odd numbers −1 and 1.
• The sum or difference of two even numbers is even, and the product of an even number with any integer is even.
• Two is the only even prime, because every other even number has 2 as a proper divisor.
• In base ten a number is even exactly when its final digit is 0, 2, 4, 6 or 8; in binary, exactly when its final bit is 0.
• The sum of the first n positive even numbers is n(n+1): 2 + 4 + 6 + 8 = 20 = 4 × 5.
• Every even perfect number has the form 2^(p−1)(2^p − 1) with 2^p − 1 prime. Euclid proved such numbers are perfect; Euler proved there are no other even ones.
• Goldbach’s conjecture — that every even number greater than 2 is a sum of two primes — has been verified by computer up to 4 × 10^18 but remains unproven.

Where they turn up

• Parity bits: serial links and parity-checked memory append one bit chosen to make the number of ones in each word even, so any single-bit flip shows up as an odd count. A single parity bit can only detect an error, not locate it; Richard Hamming generalised the idea into several overlapping checks that identify which bit moved, publishing the resulting codes at Bell Labs in 1950.
• Euler’s 1736 analysis of the seven bridges of Königsberg turns entirely on parity: a walk crossing every bridge exactly once can exist only if at most two landmasses have an odd number of bridges. All four of Königsberg’s landmasses had an odd count, so no such walk exists.
• The mutilated chessboard problem — remove two diagonally opposite corners, then try to tile the remaining 62 squares with dominoes — is impossible, because each domino covers one square of each colour while the two removed squares share a colour. Max Black posed it in his 1946 book Critical Thinking.
• Odd-even driving restrictions ration road access by the parity of a licence plate. Beijing used the scheme around the 2008 Olympics, Paris during pollution episodes, and Delhi ran a high-profile trial in January 2016.
• House numbering in much of the English-speaking world puts even numbers on one side of a street and odd on the other. This is a municipal convention rather than a rule, and cities disagree about which side gets which.