First 20 odd numbers

Odd numbers carry the older and more charged half of the parity distinction. The Greek perissos meant excessive or left over — the unit that refuses to be halved — and Euclid's Elements, assembled at Alexandria around 300 BCE, defines an odd number twice over in Book VII: as one not divisible into two equal parts, and as one differing from an even number by a unit.

Odd numbers were also the Greek route to square numbers. Laid out as pebbles, successive odd numbers form L-shaped borders — gnomons — around a growing square, and adding them keeps the figure square: 1, then 1+3 = 4, then 1+3+5 = 9, and so on to n². Nicomachus of Gerasa set this out around 100 CE in the Introduction to Arithmetic, though the figure is almost certainly older, part of the pebble arithmetic the Pythagoreans are said to have practised in southern Italy in the fifth century BCE.

The sharpest classical use of parity is a proof. Aristotle, in the Prior Analytics, refers to the demonstration that a square's diagonal is incommensurable with its side, carried out by showing that otherwise odd numbers would have to equal even ones. A version of that argument was later attached to Book X of the Elements; modern editions print it in an appendix and treat it as a subsequent addition rather than Euclid's own work. The familiar story that the Pythagorean Hippasus was drowned at sea for divulging the result comes from Pappus and Iamblichus, writing some seven centuries after the supposed events, and is generally read as legend.

Virgil supplied the cultural tag line in his eighth Eclogue, from a collection composed somewhere between roughly 44 and 38 BCE: numero deus impare gaudet, the god delights in an odd number. A preference for odd numbers runs through Roman and later European custom, and florists still bundle stems in threes and fives — though whether today's habit descends from the ancient one, or simply resembles it, is not something the record settles.

Key properties

• An integer is odd exactly when dividing it by 2 leaves a remainder of 1 — when it can be written as 2k + 1 for some integer k.
• The sum of two odd numbers is even, the sum of an odd and an even number is odd, and the product of two odd numbers is odd.
• The sum of the first n odd numbers is exactly n²: 1 + 3 + 5 + 7 = 16 = 4².
• Every odd number is the difference of two consecutive squares: 2n + 1 = (n+1)² − n².
• In base ten a number is odd exactly when its final digit is 1, 3, 5, 7 or 9; in binary, exactly when its final bit is 1.
• An integer and its square always share a parity, so every odd square is the square of an odd number.
• No odd perfect number has ever been found, and whether one exists is still an open problem; every perfect number known is even.
• The weak Goldbach conjecture — every odd number greater than 5 is a sum of three primes — was given a proof by Harald Helfgott, announced in 2013 and broadly accepted.