Mathematical Curiosities

In 1964 C. P. Willans wrote down a single closed-form expression that returns the n-th prime number exactly. It leans on Wilson's theorem to spot primes and stacks floor functions and a cosine to count them. The formula is perfectly exact — and gloriously useless to compute — but it settles a quiet question: yes, such a formula exists.

Deep in the decimal expansion of π, at the 762nd place, six nines appear in a row: 999999. The run is named after Richard Feynman, who joked that he would love to memorize π all the way to that spot — so he could rattle off the digits and finish with "…nine, nine, nine, nine, nine, nine, and so on," cheekily hinting that π might just be rational after all.

Write down the first 38 digits of π — starting with the 3 and continuing through the decimals — and you get a 38-digit integer. Remarkably, that integer is prime. It looks like the opening of π, yet it stands on its own as a prime number, divisible by nothing but 1 and itself.

Walk along the Fibonacci numbers as far as you like and you will never again land on a perfect square once you pass 144. Beyond the trivial 1, it is the only large Fibonacci number that is also a square — a fact proven by John H. E. Cohn in 1964. Fittingly, 144 sits at position twelve and equals twelve squared.

Add up the numbers in the n-th row of Pascal's triangle and you get 2ⁿ — a familiar sight. But take the product of those same numbers, call it tₙ, and something stranger appears. Harlan Brothers found that a simple ratio of consecutive row products converges to e, the base of the natural logarithm, quietly tucked inside a triangle of binomial coefficients.

A square root can describe itself. The identity below contains √x on both sides, so substituting it back into its own denominator over and over unfolds into a generalized continued fraction. The recursion converges to √x for any x — a tidy, self-referential way to compute roots by hand.

Here is a quirk that lives in language rather than arithmetic: in English, the name of every single odd number contains the letter e. One, three, five, seven, nine — keep going as far as you like and the e never disappears. Even numbers make no such promise; two, four, and six get by just fine without it.

Only six positive integers share this curious property: each is a perfect cube whose digits, added up, bring you right back to the cube's root. They are 1, 8, 17, 18, 26, and 27 — and no seventh such number exists. They're named after Henry Dudeney, the celebrated English puzzle-maker who discovered them.

Take a number made only of ones — a repunit — and square it. Out tumbles a perfect palindrome that climbs up to the count of ones and then walks right back down. The staircase holds beautifully up to nine ones; beyond that the digits begin to carry and the mirror finally cracks.

Line up the counting numbers in order, multiply by nine, then add the next number along — and the answer comes out as a string of pure ones. Each rung of the ladder stretches a digit longer than the one above it, marching neatly from 11 all the way to 111111111.

Paul Erdős was so prolific — more than 1,500 papers with over 500 collaborators — that mathematicians measure their closeness to him with an Erdős number. Erdős himself is 0. Anyone who co-authored a paper with him is a 1. Co-author with one of them and you're a 2, and so on down the chain. Most working mathematicians land somewhere between 2 and 5, and even a few scientists outside mathematics can claim a finite number.

Raise one number to another and then swap their roles, and you'll almost always get two different answers. Almost. A rare few pairs are perfectly indifferent to the switch — the whole-number classic is 2 to the 4th equalling 4 squared. Hidden among the fractions lives a subtler partner, where two lopsided fractions trade places and the equation still balances to the last decimal.

Raise each digit of a number to a fixed power and add them up — and sometimes you land right back on the number you started with. These self-admiring integers are called narcissistic numbers. Among three-digit numbers there are only four: 153, 370, 371, and 407. No fifth exists. The pattern continues at four, five, and seven digits — each digit raised to the matching power, summing back to the number itself.

Narcissistic numbers are not confined to three digits. Take 115132219018763992565095597973971522401 — a 39-digit integer — raise each of its digits to the 39th power, and add; the total returns the number you started with. The identity below is long, but every digit on the left reappears on the right, elevated to the same power.

Raise each digit of 2178 to the fourth power and sum — you land on 6514. Raise each digit of 6514 to the fourth power and sum — you land back on 2178. The two numbers chase each other in an endless loop, each one hiding inside the other's digits.

In a narcissistic number, every digit is raised to the same power. Disarium numbers play a subtler game: the first digit is raised to the first power, the second to the second, the third to the third — and the sum rebuilds the original number. Among three-digit integers, only four qualify.

Read the digits of 2592 straight across — 2, 5, 9, 2 — and they quietly spell out the very sum that builds the number: two to the fifth, times nine squared. Numbers that can be rebuilt from their own digits this way are called Friedman numbers, and 2592 is one of the tidiest of them all.

In 1772 Leonhard Euler noticed that a plain little quadratic spits out prime after prime. Feed it every integer from 0 up to 40 and each result is prime — forty-one primes marching in a row. But no simple formula can churn out primes forever: at n = 41 the expression collapses to 41 squared, and the streak finally breaks.

Stack the digit 3 on itself — 31, then 331, then 3331 — and something uncanny happens. The first seven numbers in the ladder are all prime. Keep going and the streak looks unstoppable. Then, on the eighth step, 333333331 quietly factors as 17 times 19,607,843, and the run is over.

The sixteen-digit number 0588235294117647 is the repeating block of the fraction 1/17, and it hides a startling habit. Multiply it by 2, by 3, by any number up to 16, and every single product is built from those exact same sixteen digits — merely rotated to a new starting point. Numbers with this property are called cyclic numbers.

Square 588 and square 2353, then add the results — and the answer reads as if the two numbers were simply glued end to end: 5882353. It is a rare coincidence, but a striking one: two separate squares, one sum, and out comes a single number built from both originals.

The number 37 is stitched together from the digits 3 and 7 in more ways than one. Square them, subtract their product, and you get 37. Cube them, add, divide by their sum — and 37 appears again. Two different recipes, one number.

Add up the digits of 4913 and you get 17 — cube that, and you land right back on 4913. The very same trick works for 19683, whose digits sum to 27. Take the digits of the number, add them, cube the total, and the original number reappears as if by sleight of hand.

Cube each digit and add — most numbers eventually settle down. But 919 and 1459 chase each other forever: cube the digits of 919 and you get 1459; cube the digits of 1459 and you land back on 919. Neither equals its own digit-cube sum, yet together they form a closed loop of length two.

Take a run of consecutive integers, split them in two, square every member on each side, and add — sometimes both totals match. With 10 through 14, three squares on the left equal two on the right, both summing to 365. Stretch the run to 21 through 27 and the pattern holds again: four consecutive squares balance three, landing on 2030.

The most famous Pythagorean triple lines up three consecutive numbers: 3, 4, and 5. Climb one dimension higher and the pattern echoes with four consecutive numbers — 3, 4, 5, and 6 — this time as cubes. Two neat little coincidences sitting right next to each other on the number line.

In the Book of Revelation, 666 is called the number of the beast — and the fear of it even has a name: hexakosioi­hexekonta­hexaphobia. But strip away the folklore and 666 turns out to be one of the most playful integers in mathematics. It is the sum of the squares of the first seven primes. It can be built from the digits 1 through 9 in several different ways. It even appears as a palindromic run of cubes, climbing from 1 to 6 and back down again. The superstition runs deep enough that in 1989 Ronald and Nancy Reagan had their Bel-Air address changed from 666 St. Cloud Road to 668.

Take the number 2646798 and read off its digits: 2, 6, 4, 6, 7, 9, 8. Now raise each digit to a power that climbs from one to seven, in order. Add the results and you land right back on 2646798 — the very number you started with. It is a pure numerical coincidence, but a delightfully tidy one.

Look closely at the universal recycling symbol — three arrows chasing one another in a triangle — and you are looking at a Möbius strip. One of the arrows carries a half-twist, the same twist that turns a strip of paper into a surface with only one side and one edge. Gary Anderson designed it in 1970 for a college competition on the first Earth Day, choosing a form with no beginning and no end to echo endless reuse.

Some numbers describe themselves. In 6210001000, the first digit tells you how many zeros appear; the second tells you how many ones; the third how many twos — and so on through all ten digits. Read the list below and every line checks out. It is one of the few ten-digit numbers that pulls off this trick.

Multiply a number by its own reverse — 861 times 168, say — and you may expect a one-off coincidence. But some products refuse to stay unique. They can be assembled from a second pair of reversed factors as well: 144,648 equals both 861 × 168 and 492 × 294. Numbers with this double identity are called EPRNs — Equal Products of Reversible Numbers. They are uncommon, but the pattern keeps turning up as the digits grow longer.

A limerick is a five-line poem with a strict rhyme scheme. This one, collected by Betsy Devine and Joel E. Cohen, is unusual: read aloud, its syllables spell out a genuine integral identity. Both sides evaluate to exactly one-third — the joke lands because every phrase in the poem corresponds to a symbol in the equation.

At first glance, 35 − 3² − 5² and 75 − 7² − 5² look like unrelated subtractions. Work them out and both sides collapse to exactly 1. The digits echo each other — 35 and 75, 3 and 7 flanking a shared 5² — yet there is no deeper theorem hiding behind the symmetry. It is a tidy arithmetic coincidence, the kind that makes you double-check your mental math.

Multiply 73, 9, and 42 together and you get 27,594. Now take the very same digits — 7, 3, 9, 4, and 2 — and regroup them as 7 and 3942. Their product is still 27,594. The factors look completely different, yet the digit set is identical and the answer does not budge.

Square 12 and you get 144. Reverse the digits of 12 to get 21, square that, and reverse the result — you get 441, the reverse of 144. The same trick works for 13 and 31. It is not magic: when (10a + b)² expands with no carries, reversing the digits of the number reverses the digits of its square.

Take the digits 1 through 9 in descending order and subtract them in ascending order — 987654321 minus 123456789. The answer is 864197532. Remarkably, all three numbers — minuend, subtrahend, and difference — are pandigital: each uses every digit from 1 to 9 exactly once.

Pick any four-digit number with at least two distinct digits — say 2736. Arrange its digits to form the largest and smallest numbers possible (7632 and 2367), then subtract. Repeat the same operation on the result. No matter where you start, within seven steps you always arrive at 6174 and stay there. For three-digit numbers the same process always leads to 495.

Some primes survive being trimmed from one end. The number 739397 survives being trimmed from both. Peel digits off its right side — 739397, 73939, 7393, 739, 73, 7 — and every step stays prime. Build it up instead by adding digits to its left — 7, 97, 397, 9397, 39397, 739397 — and every step is prime again. Numbers with this double property are called two-sided primes, and only fifteen of them exist in base ten. 739397 is the largest — and, it has been proven, the last.

Some primes keep their status no matter how many digits you strip away. A right-truncatable prime stays prime when you lop off digits from the right — 73939133, then 7393913, then 739391, all the way down to 7. Left-truncatable primes work from the other end: start with 357686312646216567629137 and delete from the front, and every intermediate number is still prime. In base ten, the longest right-truncatable prime has eight digits; the longest left-truncatable prime has twenty-four.

In 1955 Pedro A. Pisa published an equation that refuses to die. Three numbers on the left, three on the right — add them and both sides match. Then square every term and both sides still match. Peel digits from the ends, row by row, and the balance holds all the way down to single digits and back up to six. Hack away at any column in sync and the truth survives.

Write a three-digit number twice in a row and you have secretly multiplied by 1001. Factor 1001 into 7, 11, and 13 — and the six-digit giant collapses back to where it started, one clean division at a time.

The sets {1, 6, 7, 17, 18, 23} and {2, 3, 11, 13, 21, 22} look unrelated, yet they are arithmetically twins. Add the numbers in each set: equal. Square every number and add: still equal. Cube them: equal. Raise to the fourth power: equal. Fifth power: equal. Through every power from 1 to 5, the two sides remain in perfect balance.

The sets {1, 5, 10, 18, 23, 27} and {2, 3, 13, 15, 25, 26} strike the same rare balance. Add them — equal. Square, cube, raise to the fourth, raise to the fifth — equal every time. Identities like this are called multigrade equations, and matching two six-number sets through five separate powers is a hard-won piece of construction, not a coincidence.

Start with 9, keep prepending the next lower digit, multiply by 9, and adjust the addend down by 1 each time. Every result is a string of 8s — and the pattern keeps going even when the addend turns negative.

Circle April 4, June 6, August 8, October 10, and December 12 on any calendar you like, and all five dates land on the exact same day of the week — true in leap years and ordinary ones alike, every single year. In 2026, for instance, that shared day is a Saturday. It is the visible edge of what the mathematician John Conway called the Doomsday rule: a small set of dates that always share a year's “doomsday,” letting anyone work out the weekday of any other date in their head. The same trick reaches further still — 5/9, 9/5, 7/11, and 11/7 land on it too, along with the last day of February.

Multiply 8,589,934,592 by 116,415,321,826,934,814,453,125 and every messy digit vanishes — the product is a flawless 1 followed by thirty-three zeros. The trick hides in the factors: the first number is 2 raised to the 33rd power, the second is 5 raised to the 33rd power. Since 2 and 5 are exactly the prime factors of 10, raising both to the same power and multiplying always collapses into a power of ten, however sprawling the numbers look along the way.

Add the squares of 3 through 9, then divide by the sum of the squares of 1 through 7 — two runs of seven consecutive integers, offset by two — and the fraction reduces to precisely 2, no remainder. Slide the same seven-term window anywhere else along the number line and the tidy 2 doesn't come back; it is a coincidence of these two particular runs, not a general rule.

The Fibonacci sequence grows by the plainest rule imaginable: each term is the sum of the two before it — 0, 1, 1, 2, 3, 5, 8, 13, 21, and onward forever. Buried in that simple sum is an unexpected shortcut. Read off two neighboring terms — 5 and 8, 8 and 13, 13 and 21 — and the smaller is, to within about half a percent, how many miles equal the larger in kilometers. It works because consecutive Fibonacci numbers divide into an ever-closer approximation of the golden ratio, φ ≈ 1.618, and a mile happens to equal almost exactly 1.609 kilometers — two unrelated constants sitting a hair's width apart.

Pair each base from 1 to 9 with a different exponent from 1 to 9, then add the nine resulting powers together — shuffle the pairing, and the total should change every time. Remarkably, at least three separate permutations refuse to budge from the exact same sum: 134,757,431. Swap which power lands on which base, and the number holds steady.

Raise each digit of a number not to a power but to a factorial, and add the results — for two special numbers, the sum lands exactly back on where it started. These are called factorions. Past the trivial cases 1 and 2, where a lone digit's factorial equals the digit itself, only two others have ever been found: 145 and 40585. No third has ever turned up, and none ever will — since 9! is 362,880, once a number reaches eight digits its digit-factorial sum can no longer keep pace with it.

Add up the first n odd numbers, one at a time, and the running total always lands on n² — a perfect square, every step of the way. Group the odd numbers differently instead — into blocks that grow by one term each time, 1, then 3+5, then 7+9+11 — and the block totals climb through the cubes just as cleanly. Two staircases, built from nothing but the odd numbers in order.

The digits 1 through 9 in ascending order and in descending order both fold neatly out of the same hidden number: 100. Tuck a 100 together from small arithmetic — 86 plus 2 times 7, or 97 plus 6 over 2 — raise it to the fifth power, nudge it with a stray add or subtract, then divide by 3⁴, and out comes 123456789 on one side, 987654321 on the other.

Multiply 54981 by 62037 and the ten digits scattered across both factors — 0 through 9, each used exactly once — simply reshuffle themselves into the product, 3410856297. Square that product and the trick repeats one level up: the twenty digits of the answer are once again 0 through 9, this time with every digit turning up exactly twice.