पहले 500 सम संख्याएँ

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.

मुख्य गुण

• 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.