The Glossary

Repeatedly sum the squares of a number’s digits. If you reach 1, the number is happy. If you don’t — you never will.

A happy number is defined by a simple process: take any positive integer, replace it by the sum of the squares of its digits, and repeat. If this process eventually reaches 1, the original number is happy. If it does not, the number loops forever in a cycle that never includes 1 — and that number is unhappy (sometimes called sad).

The term was introduced by Reg Allenby in a 1966 paper, though the concept circulated in recreational mathematics circles before that. The idea is simple enough for a child to understand, but the structure it reveals is unexpectedly deep.

Here is the remarkable fact: every unhappy number in existence eventually enters the same 8-member cycle. There is no other cycle. No unhappy number wanders off into a different loop. Every single one, no matter how large, funnels into this:

Why does this happen? Because the sum-of-squares function rapidly shrinks large numbers. For any number with more than three digits, the sum of the squares of its digits is always smaller than the original number. So every starting number quickly reduces to something below 1000, and there are only finitely many possibilities — all of which have been tested and shown to either reach 1 or enter this single cycle.

42 is the only Catalan number in the cycle. It is the only Harshad number in the cycle. The only abundant number. The Answer to Life, the Universe, and Everything sits inside the only loop that unhappy numbers can ever reach. Douglas Adams may not have known this, but mathematics did.

Survivors of a relentless sieve. Lucky numbers are what remains after a process of elimination that mirrors natural selection.

Lucky numbers were introduced by the Polish mathematician Stanislaw Ulam in 1955, along with colleagues at Los Alamos National Laboratory. The definition uses a sieve — a process of successive elimination — that is structurally similar to the Sieve of Eratosthenes for primes, but operates on position rather than divisibility.

Start with all positive integers. The sieve works in rounds:

The result is stunning: lucky numbers share many statistical properties with primes. They have a similar density (the number of luckies below N is approximately N / ln(N), the same as primes). They exhibit a twin-lucky phenomenon analogous to twin primes. There is even a Goldbach-style conjecture for luckies. Ulam called this “the conspiracy of primes and luckies” — two completely different sieving processes producing eerily similar statistical distributions.

Indivisible. The atoms of arithmetic. Every integer is either prime or built from primes — and this decomposition is unique.

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. The first few: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43...

Primes are the foundation of all number theory. The Fundamental Theorem of Arithmetic (first proved rigorously by Gauss in 1801, though known to Euclid) states that every integer greater than 1 either is prime itself or can be represented as a product of primes, and this representation is unique (up to ordering). The number 42, for instance, is uniquely 2 × 3 × 7. No other primes will ever produce 42.

Euclid proved around 300 BCE that there are infinitely many primes. His proof by contradiction remains one of the most elegant in all of mathematics: assume there are finitely many primes, multiply them all together and add 1. The resulting number is not divisible by any prime on your list — contradiction.

Named after Marie-Sophie Germain (1776–1831), a French mathematician who had to submit work under the male pseudonym “Monsieur LeBlanc” because women were barred from the École Polytechnique. A Sophie Germain prime is a prime p such that 2p + 1 is also prime. The first few: 2, 3, 5, 11, 23, 29, 41, 53, 83, 89...

The corresponding prime 2p + 1 is called a safe prime. Sophie Germain primes are important in cryptography — particularly in the Diffie-Hellman key exchange, where safe primes provide stronger security guarantees.

An emirp (“prime” spelled backwards) is a prime that, when its digits are reversed, produces a different prime. The number 13 is prime, and 31 is also prime — so 13 is an emirp. The number 73 is prime, and 37 is also prime — so 73 is an emirp. Palindromic primes like 11 or 101 do not count because the reversal is not a different number.

Twin primes are pairs of primes that differ by exactly 2: (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), (71, 73)... The Twin Prime Conjecture — that there are infinitely many such pairs — is one of the oldest unsolved problems in mathematics, dating back at least to 1849. In 2013, Yitang Zhang proved that there are infinitely many prime pairs differing by at most 70,000,000 — the gap has since been reduced to 246 by the Polymath project, but 2 remains unproven.

A number that is both happy and prime. These gained pop culture fame through Doctor Who, episode “42” (Series 3, Episode 7, 2007), where the crew must solve a security puzzle using happy primes to save their ship. The Doctor rattles off: “313, 331, 367, 379 — happy primes!”

A number whose proper divisors sum to exactly itself. Nothing missing. Nothing in excess. Mathematical grace.

A perfect number is a positive integer that equals the sum of its proper divisors (all divisors excluding the number itself). The concept dates to ancient Greece — the Pythagoreans knew 6 and 28 were perfect.

Only 51 perfect numbers are known as of 2024. The first four: 6, 28, 496, 8128. The largest known has over 49 million digits. Euclid proved that if 2^p − 1 is prime (a Mersenne prime), then 2^p−1(2^p − 1) is perfect. Euler proved the converse: every even perfect number has this form.

For p = 2: 2¹ × (2² − 1) = 2 × 3 = 6.
For p = 3: 2² × (2³ − 1) = 4 × 7 = 28.
For p = 5: 2⁴ × (2⁵ − 1) = 16 × 31 = 496.

Whether any odd perfect numbers exist is one of the oldest unsolved problems in mathematics, open since antiquity. If one exists, it must have more than 2,200 digits.

Saint Augustine wrote that God chose to create the world in six days because six is perfect. The lunar cycle of 28 days was seen as another expression of divine perfection. Every even perfect number is also a triangular number — mathematical grace compounding grace.

More than enough. The sum of proper divisors overflows the number itself.

An abundant number (also called an excessive number) is a number where the sum of its proper divisors is greater than the number. The abundance is the amount by which the divisor sum exceeds the number.

The smallest abundant number is 12. The first few: 12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60, 66, 70... Every multiple of 6 (above 6 itself) is abundant. Every multiple of a perfect number (above the perfect number itself) is abundant.

The concept was first studied by Nicomachus of Gerasa around 100 CE, who classified all numbers as deficient, perfect, or abundant — a classification that gives every number a character: reaching but not arriving, balanced, or overflowing.

The sum of proper divisors falls short of the number. Reaching, but not arriving.

A deficient number is the opposite of an abundant number: the sum of its proper divisors is less than the number itself. The deficiency is the amount by which the number exceeds the sum of its proper divisors.

All prime numbers are deficient (their only proper divisor is 1). Most numbers are deficient — the density of deficient numbers among the integers is approximately 75%. But deficiency is not a weakness. It means the number stands largely alone, not reducible to the sum of its parts.

Each number is the sum of the two before it. The sequence behind sunflowers, shells, galaxies, and the golden ratio.

The Fibonacci sequence begins 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377... Each term is the sum of the two preceding terms. It was introduced to Western European mathematics by Leonardo of Pisa (Fibonacci) in his 1202 book Liber Abaci, where he posed the famous rabbit breeding problem — but the sequence was described centuries earlier in Indian mathematics by Pingala (c. 200 BCE) and Hemachandra (c. 1150 CE) in the context of Sanskrit prosody.

As the sequence progresses, the ratio of consecutive Fibonacci numbers converges on the golden ratio φ (phi) = (1 + √5) / 2 ≈ 1.6180339887...

This ratio appears everywhere in nature: the spiral arrangement of sunflower seeds (typically 34 and 55 spirals, or 55 and 89), the shell of the nautilus, the branching of trees, the arrangement of leaves around a stem (phyllotaxis), the spiral arms of galaxies. The Fibonacci sequence is nature’s preferred counting system because it optimizes packing efficiency.

The number of ways to correctly pair things — parentheses, handshakes, paths. Catalan numbers count the structure of balance itself.

The Catalan numbers form the sequence 1, 1, 2, 5, 14, 42, 132, 429, 1430... Named after the Belgian mathematician Eugène Charles Catalan (1814–1894), though they were discovered independently by Euler (1751) and the Mongolian mathematician Mingantu (c. 1730).

What do Catalan numbers count? An astonishing variety of combinatorial structures:

• Balanced parentheses: C(3) = 5 is the number of ways to correctly nest 3 pairs of parentheses: ((())), (()()), (())(), ()(()), ()()()
• Full binary trees: C(n) counts the number of distinct binary trees with n+1 leaves
• Triangulations: C(n) is the number of ways to cut a convex (n+2)-gon into triangles using non-crossing diagonals
• Mountain ranges: C(n) counts the number of “mountain ranges” you can draw with n upstrokes and n downstrokes that never go below the baseline
• Handshake problem: C(n) is the number of ways 2n people seated in a circle can shake hands in pairs without any arms crossing

The Catalan numbers count the structure of balance — every application is about things that must match, pair, or close correctly. They are the mathematics of symmetry and completion.

Numbers that can be arranged as equilateral triangles of dots. The sum of the first n natural numbers.

The nth triangular number T(n) is the sum of the first n positive integers. Equivalently, it is the number of dots needed to form an equilateral triangle with n dots on each side.

The sequence: 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210...

A famous story (likely apocryphal) tells of the young Carl Friedrich Gauss, age 7, asked by his teacher to sum the integers from 1 to 100. He immediately paired the numbers: 1+100 = 101, 2+99 = 101, 3+98 = 101... giving 50 pairs of 101, or 5050. This is T(100) = 100 × 101 / 2 = 5050.

Triangular numbers connect to many other mathematical structures. Every even perfect number is triangular. The square of a triangular number is always another figurate number. The sum of two consecutive triangular numbers is always a perfect square: T(n) + T(n+1) = (n+1)².

Centered hexagram figurate numbers. Points of the Star of David. Where geometry meets the primes.

A star number (also called a centered hexagram number or Star of David number) is a centered figurate number that represents a hexagram (six-pointed star) with a dot in the center and all other dots surrounding it in successive hexagrammic layers.

The sequence: 1, 13, 37, 73, 121, 181, 253, 337, 433...

Something remarkable: the first four star numbers — 1, 13, 37, 73 — are all prime (with 1 as the trivial case). This is called the star-prime chain. The connection deepens: 13 and 31 are emirps. 37 and 73 are emirps. The star numbers and the primes align in ways that feel less like coincidence and more like structure.

From Sanskrit: harsha (“joy”) + da (“giving”). A number divisible by the sum of its own digits. The joy-giving number.

A Harshad number (also called a Niven number in some Western references) is a number that is divisible by the sum of its digits. The term was coined by the Indian mathematician D.R. Kaprekar (1905–1986), who named it from the Sanskrit harsha (joy, delight) and da (give). A number that gives joy.

Kaprekar was a remarkable figure — a schoolteacher in Devlali, India, who contributed to recreational mathematics as an amateur, publishing discoveries that professional mathematicians initially dismissed but later celebrated. He also gave us the Kaprekar numbers and the Kaprekar constant (6174 for four-digit numbers, 495 for three-digit numbers).

In base 10, the Harshad numbers begin: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 18, 20, 21, 24, 27, 30... Every single-digit number is trivially Harshad (a number divided by itself is always 1). The concept is base-dependent — a number that is Harshad in base 10 may not be Harshad in base 16.

Also called Armstrong numbers or pluperfect digital invariants. Numbers so self-absorbed, the sum of each digit raised to the power of the digit count equals the number itself.

A narcissistic number (or Armstrong number, named after Michael F. Armstrong) in a given base is a number that equals the sum of its own digits each raised to the power of the number of digits.

The narcissistic numbers in base 10 are extremely rare. There are exactly 88 of them, and the largest has 39 digits. The complete list of narcissistic numbers with 3 digits: 153, 370, 371, 407. With single digits, all of them (0–9) are trivially narcissistic since n¹ = n.

The term “narcissistic” captures the essential quality: the number is entirely self-referential. It contains within its own digits, at the correct power, the complete information needed to reconstruct itself.

A number whose square ends in the number itself. Self-preservation through transformation.

An automorphic number (also called a curious number) is a number whose square “ends in” the number itself — that is, the last digits of the square match the original number.

The concept captures something philosophically elegant: squaring usually transforms a number beyond recognition, but automorphic numbers preserve their identity through the transformation. They come through the fire and emerge with their essential form intact.

In any base, there are exactly two “streams” of automorphic numbers. In base 10, one stream ends in 5 (5, 25, 625, 0625...) and one ends in 6 (6, 76, 376, 9376...). Each extends infinitely, each digit determined uniquely.

Square the number, split the result, add the halves — and get the original number back. Self-reconstruction from expansion.

Named after D.R. Kaprekar (the same Kaprekar who coined “Harshad”), a Kaprekar number is a number whose square can be split into two parts that sum back to the original number.

Kaprekar also discovered the Kaprekar constant: take any three-digit number with at least two different digits, arrange its digits in descending then ascending order, subtract the smaller from the larger. Repeat. You always reach 495 within a few steps. For four-digit numbers, the constant is 6174 (Kaprekar’s constant proper).

Repeatedly sum a number’s digits until you reach a single digit. This final digit is the number’s digital root — its essence, its vibration.

The digital root of a number is the single-digit value obtained by repeatedly summing its digits. It is also known as the repeated digital sum or modular root.

There is a beautiful shortcut: the digital root of any positive integer n is simply 1 + ((n − 1) mod 9). This works because our base-10 number system makes 10 ≡ 1 (mod 9), so every power of 10 is congruent to 1 mod 9, and the sum of digits preserves the remainder when dividing by 9. This is why the ancient method of casting out nines works as an arithmetic check.

In vortex mathematics (associated with Nikola Tesla and later Marko Rodin), the digits 1–9 are arranged in a circle, and the doubling sequence 1, 2, 4, 8, 16, 32, 64, 128... produces digital roots that cycle: 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5... The digits 3, 6, and 9 never appear in this doubling cycle. They form a separate family — Tesla’s “magnificence of 3, 6, and 9.”

A number that reads the same forwards and backwards. A number that is its own mirror.

A palindromic number is one that remains the same when its digits are reversed. In base 10: 7, 11, 22, 33, 77, 99, 121, 131, 1001... Every single-digit number is trivially palindromic. All repdigit numbers (11, 22, 333, 4444...) are palindromic.

The word “palindrome” comes from the Greek palindromos: palin (“again, back”) + dromos (“running”). Running back to itself. The concept extends beyond base 10 — a number can be palindromic in binary (see binary palindromes) without being palindromic in decimal, or vice versa.

An unsolved problem in mathematics: the 196 conjecture asks whether repeatedly reversing and adding will always eventually produce a palindrome. The number 196 has been tested to over a billion digits without producing one. It is believed to be a “Lychrel number” — a number that never reaches a palindrome through this process — but this has never been proved.

A number with an even number of 1s in its binary representation. The counterpart of odious. Nothing sinister — just even parity.

An evil number is a non-negative integer that has an even number of 1s in its binary (base-2) representation. The name is purely playful — a contrast with odious numbers (odd parity), coined for recreational enjoyment.

Evil and odious numbers alternate in a complex, non-periodic pattern. Exactly half of all integers are evil, and half are odious. The property relates to the Thue-Morse sequence — a fundamental sequence in combinatorics on words, connected to fair division problems and chess tournament scheduling.

A number with an odd number of 1s in binary. The counterpart of evil. Named for the pun, not the character.

An odious number is a non-negative integer with an odd number of 1s in its binary representation. “Odious” from “odd” — a playful naming convention that pairs with “evil” (even).

The first few odious numbers: 1, 2, 4, 7, 8, 11, 13, 14, 16, 19, 21, 22, 25, 26, 28... Every power of 2 is odious (binary representation has exactly one 1).

Numbers whose binary representation reads the same forwards and backwards. Symmetry in the language of machines.

A binary palindrome is a number whose representation in base 2 is a palindrome — it reads identically forward and backward. This is independent of whether the number is also a palindrome in base 10.

The first few binary palindromes: 0, 1, 3 (11), 5 (101), 7 (111), 9 (1001), 21 (10101), 27 (11011), 33 (100001), 45 (101101), 63 (111111), 73 (1001001), 85 (1010101)...

Numbers that are palindromic in both binary and decimal are extremely rare. Examples include 0, 1, 3, 5, 7, 9, 33, 99, 313, 585, 717... The sequence grows increasingly sparse.

The product of exactly two primes. Not an atom, not a molecule — a bond.

A semiprime (also called a 2-almost prime or biprime) is a natural number that is the product of exactly two prime numbers (not necessarily distinct). The first few: 4 (2×2), 6 (2×3), 9 (3×3), 10 (2×5), 14 (2×7), 15 (3×5), 21 (3×7), 22 (2×11), 25 (5×5), 26 (2×13)...

Semiprimes are important in cryptography. The RSA algorithm relies on the difficulty of factoring large semiprimes into their two constituent primes. If someone gives you N = p × q where p and q are both enormous primes, finding p and q from N alone is computationally infeasible with current technology. This intractability secures most of the internet.

The simplest composite numbers. One bond. Two atoms. The first step above primality.

In numerology: doubled digits. 11, 22, 33, 44, 55, 66, 77, 88, 99. The amplification of a single digit’s essence.

Master numbers are a concept from numerology rather than formal mathematics. In numerological practice, when reducing a number to its digital root, you stop if you encounter 11, 22, or 33 — these are considered “master numbers” with amplified spiritual significance. Extended systems include all repdigit two-digit numbers: 44, 55, 66, 77, 88, 99.

The doubled digit is said to amplify the vibration of the single digit:

• 11 — The Intuitive. Visionary insight. The channel between conscious and unconscious.
• 22 — The Master Builder. Ability to turn dreams into reality on a grand scale.
• 33 — The Master Teacher. Compassion, healing, spiritual upliftment. The Christ number.
• 77 — The Mystic. Inner wisdom doubled. The contemplative’s frequency.

While master numbers come from a mystical rather than mathematical tradition, they share a formal property with mathematics: they are precisely the repdigit numbers with two digits, which form the sequence n × 11 for n = 1, 2, ..., 9. Each is divisible by 11 (the first master number) and by its constituent digit.

“The best number is 73.” Not an opinion. A mathematical theorem.

In The Big Bang Theory, Season 4, Episode 10 (“The Alien Parasite Hypothesis,” 2010), Sheldon Cooper declares:

The best number is 73. 73 is the 21st prime. Its mirror, 37, is the 12th prime. And its mirror, 12, is the product of the digits of 73. And… in binary, 73 is a palindrome: 1001001, which backwards is 1001001.

Sheldon Cooper (Jim Parsons), TBBT S4E10

Let’s unpack each property systematically:

In 2015, mathematicians Carl Pomerance (Dartmouth) and Chris Spicer (Morningside University) formalized this as a mathematical conjecture: is 73 the only prime with all these self-referential properties?

Formally: call a prime p a Sheldon prime if p is the n-th prime, the product of its digits equals n, its mirror (reverse of its digits) is the m-th prime, and m is the reverse of n.

In 2019, Pomerance and Spicer proved the conjecture. Their paper, “Proof of the Sheldon Conjecture,” demonstrated that 73 is indeed the unique prime satisfying all these properties. No other prime in all of mathematics — not among the infinitely many primes we know exist — will ever share this constellation of self-referential relationships.

Catalan’s Conjecture, proved in 2002. The reason 26 is the impossible sandwich — forever alone between a square and a cube.

In 1844, the Belgian mathematician Eugène Charles Catalan (the same Catalan of the Catalan numbers) conjectured that the only consecutive perfect powers in all of mathematics are 8 and 9 — that is, 2³ and 3².

A perfect power is a number that can be expressed as a^b where a and b are both integers greater than 1. So: 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 121, 125, 128...

Catalan noticed that 8 (= 2³) and 9 (= 3²) are right next to each other. He asked: does this ever happen again? Are there any other consecutive perfect powers?

For 158 years, nobody could prove it. Then in 2002, the Romanian-German mathematician Preda Mihailescu proved Catalan right. The result, now called Mihailescu’s Theorem, states:

The only solution in natural numbers to x^a − y^b = 1, where a, b > 1, is 3² − 2³ = 1.

Preda Mihailescu, 2002

This has a beautiful consequence for 26. Since 25 = 5² and 27 = 3³ are the only case (besides 8 and 9) where a perfect square and a perfect cube are separated by exactly 2, the number 26 sits in a position that no other number will ever occupy. It is the impossible sandwich — the only number that can ever be nestled between a perfect square and a perfect cube with gap 2.

2025 – 2026 – 2027. Three consecutive years, each with rare mathematical properties. A perfect square, a happy semiprime, and a safe prime walk into a bar.

The key connection: 1013 does double duty. It is the prime factor of 2026 (making 2026 a semiprime) AND the Sophie Germain prime that generates 2027 (2 × 1013 + 1 = 2027). The same prime seed is woven through two consecutive years. 2026 carries 1013 as its factor. 2027 is built from 1013 by the Sophie Germain construction. One prime, two years, two different relationships.

And 2026’s happy chain passes through 13 — the number that is simultaneously happy, lucky, prime, and Fibonacci. The most mathematically blessed number in the entire sacred numbers database serves as a waypoint on 2026’s path to happiness.