Where even numbers fear to tread 🔢
At the very beginning of mathematical history is the Lebombo bone from the border of South Africa and Swaziland, dating to around 35,000-43,000 years ago. It has 29 notches carved in a row - potentially the earliest known tally stick, predating written language by about 28,000 years!
This means humans were thinking about numbers - including counting in odd and even quantities - for tens of thousands of years before we invented writing. It's our oldest evidence that humans were keeping track of quantities, possibly lunar cycles, or other numerical patterns in their world.
Long before Pythagoras, long before Babylon, there was the Ishango bone - discovered in 1960 near Lake Edward in the Democratic Republic of Congo. This baboon fibula, carved with deliberate notches arranged in three columns, dates back approximately 20,000-25,000 years, making it one of the most sophisticated early mathematical artifacts!
What makes it mind-blowing:
Belgian anthropologist Jean de Heinzelin de Braucourt, who discovered the bone, noted that the markings were "deliberately planned" - not random scratches but organized mathematical thinking from our African ancestors in the Stone Age.
Fast forward to around 3000-2000 BC, and the Sumerians and Babylonians in ancient Mesopotamia (modern-day Iraq) developed sophisticated number systems carved into clay tablets. With hundreds of tablets surviving, we know they:
The Babylonians didn't just count - they did algebra, geometry, and even calculated square roots. A tablet from 1600 BC shows √2 calculated to 7 decimal places!
The Egyptians (from before 3000 BC) used hieroglyphic symbols for different powers of 10. While they didn't have a specific symbol for "odd," their mathematical papyri show they understood that numbers ending in 1, 3, 5, 7, or 9 behaved differently than those ending in 0, 2, 4, 6, or 8.
The Rhind Mathematical Papyrus (c. 1650 BC) contains problems that implicitly use odd and even number properties, particularly in fraction calculations.
Around 500 BC, Pythagoras and his followers didn't just discover that whole triangle thing. They also developed a philosophical system where numbers had mystical properties. Odd numbers were considered "unlimited" and even numbers "limited," because odd numbers cannot be divided into two equal parts - they always have that extra unit that refuses to be paired.
The Pythagoreans believed that understanding numbers was the key to understanding the universe itself. They saw mathematical patterns everywhere in nature and considered number theory a path to enlightenment! ✨
Ancient Greeks considered 1 to be the "even-odd" (ἀρτιοπέριττον) - neither fully even nor fully odd. Their logic? Adding 1 to an even number makes it odd, and adding 1 to an odd number makes it even. Therefore, 1 must be both!
This confusion persisted for millennia. Even in the 19th century, educator Friedrich Wilhelm August Fröbel taught students that 1 was neither even nor odd, attaching deep philosophical meaning to this mathematical identity crisis.
In 1742, Christian Goldbach wrote to Euler proposing that every even number greater than 2 is the sum of two prime numbers. For example:
The twist? It's been verified up to 4 × 1018 but never proven for ALL even numbers. The "weak" version (every odd > 5 is the sum of three primes) was only proven in 2013 by Harald Helfgott!
Pierre de Fermat claimed (in a margin, because of course) that xn + yn = zn has no positive integer solutions for n > 2.
The kicker? To prove it for all n, you only need to prove it for odd primes! Why? Because every integer greater than 2 is either divisible by 4 or by an odd prime. Fermat himself proved n=4, so mathematicians spent 358 years working on odd prime cases until Andrew Wiles cracked it in 1995.
Sophie Germain (the only woman in our odd tale) proved the first case for all primes less than 270. Not bad for someone who had to pretend to be a man to study math! 💪
This 1963 theorem states that every finite group of odd order is solvable. The proof? A casual 255 pages. The implication? Deep connections between odd numbers and abstract algebra that most mathematicians still don't fully understand.
It's proof that odd numbers aren't just for counting - they're fundamental to the structure of mathematics itself!
A perfect number equals the sum of its proper divisors. For example, 6 = 1 + 2 + 3. We know the formula for even perfect numbers (discovered by Euclid around 300 BC), but odd perfect numbers?
Status: Unknown for 2,300+ years 🤷
What we DO know:
They're the Sasquatch of mathematics - rumored, elusive, and stubbornly resistant to selfies. 📸
Among Trekkies, there's a legendary rule: every odd-numbered Star Trek movie is terrible. British sitcom Spaced (co-written by Simon Pegg) famously declared: "Sure as day follows night, sure as eggs is eggs, sure as every odd-numbered Star Trek movie is sh*t."
The evidence:
The irony? Simon Pegg later wrote Star Trek Beyond, film #13. Did he break the curse or prove it? The debate rages on!
One, three, five, seven, nine, eleven, thirteen... Seriously, try to find an odd number without the letter "e" in English. We'll wait.
Add up the first n odd numbers, and you always get n2:
This is called Nicomachus's Theorem, and it's beautifully geometric: arrange dots in L-shapes and watch squares form!
The year 2025 is oddly special:
The ancient Babylonians (around 1800 BC) had clay tablets with tables of odd and even numbers! Their sexagesimal (base-60) system was so sophisticated that they could distinguish between numbers divisible by 2 and those that weren't - crucial for their complex calculations involving fractions and astronomy.
This legacy lives on today: we still use 60 seconds in a minute and 60 minutes in an hour, thanks to the Babylonians' love of highly divisible numbers!
Every prime number is odd... except for the number 2, which is the "oddest even number" - it's the only even prime! This makes 2 special, unique, and the ultimate party crasher in the prime number club.
Addition:
Multiplication:
Any two consecutive odd numbers (like 13 and 15, or 99 and 101) have a highest common factor of 1. They're mathematical loners, sharing nothing but their oddness!
Beyond their quirky properties, odd numbers are fundamental to:
RSA encryption (the tech securing your online banking) relies on the difficulty of factoring large numbers into odd primes. Every time you shop online, you're trusting in the oddness!
Fermions (particles like electrons) obey Fermi-Dirac statistics, which depend on odd/even symmetry properties. Literally, the universe cares about oddness at the quantum level!
From five-petaled flowers to the three-fold symmetry of certain crystals, odd numbers appear throughout nature. The Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, 21...) contains infinitely many odds, creating the golden ratio that defines beauty itself!
If you enjoyed this odd journey, dive deeper:
Earliest Mathematical Evidence:
• Wikipedia: Ishango Bone - The 20,000-year-old African mathematical artifact • Mathematical Association of America: Mathematical Treasure - Ishango Bone • The Accomplish Magazine: Ancient African Mathematics • Wikipedia: Babylonian Mathematics • MacTutor History of Mathematics: Babylonian MathematicsHistorical Sources:
• Wikipedia: Parity (Mathematics) - The definitive odd/even encyclopedia entry • Origins of Mathematics: Even and Odd - Ancient Greek perspectivesGoldbach's Conjecture:
• Wikipedia: Goldbach's Conjecture • Wolfram MathWorld: Goldbach Conjecture • Plus Magazine: Mathematical Mysteries of GoldbachPerfect Numbers:
• Commonplace Fun Facts: The Search for Odd Perfect Numbers • Medium: The Oldest Unsolved Problem in MathFermat's Last Theorem:
• Wikipedia: Fermat's Last Theorem • Wikipedia: Proof for Specific ExponentsPop Culture:
• Screen Rant: Star Trek's Odd-Numbered Movie Curse Explained • StarTrek.com: 10 Unexpected Star Trek References • TV Tropes: Star Trek Movie CurseFun Facts:
• The Aperiodical: Numerical Coincidences for 2025 • Science Explorer: 6 Odd Facts About Numbers • SplashLearn: Odd Numbers Definition & Examples