Book:David Wells/Curious and Interesting Numbers/Second Edition/Errata

Errata for 1986: David Wells: Curious and Interesting Numbers (2nd ed.)

Absolutely Normal Number

$0 \cdotp 12345 \, 67891 \, 01112 \, 13141 \, 51617 \, 18192 \, 02122 \ldots$:

... [ The Champernowne constant ] is also normal, that is, whether expressed in base $10$, or any other base, each digit occurs in the long run with equal frequency.

Historical Note on Doubling the Cube

$1 \cdotp 25992 \, 10498 \, 94873 \, 16476 \ldots$:

The legend was told that the Athenians send a deputation to the oracle at Delos to inquire how they might save themselves from a plague that was ravaging the city. They were instructed to double the size of the altar of Apollo.

Brun's Constant

$1 \cdotp 90216 \, 054 \ldots$:

Brun's constant ... Its value is exceedingly hard to calculate. The best estimate is $1 \cdotp 90195 \pm 10^{-5}$.

Positive Integer is Sum of Consecutive Positive Integers iff not Power of $2$

$2$:

A positive integer is the sum of two or more consecutive integers if and only if it is not a power of $2$.

Tamref's Last Theorem

$2$:

Fermat's equation being exceedingly difficult to solve, several mathematicians have noticed in an idle moment that $n^x + n^y = n^z$ is much easier. Its only solutions in integers are when $n = 2$, and $2^1 + 2^1 = 2^2$.

$3^x + 4^y$ equals $5^z$ has Unique Solution

$2$:

Also, the only solution of $3^x + 4^y = 5^z$ in integers is $x = y = z = 2$. The equation $5^x + 12^y = 13^z$ has the same unique solution.

Continued Square Root of $1, 2, 3, 4, \ldots$

$3$:

The value of the infinite nested root, $\sqrt {1 + \sqrt {2 + \sqrt {3 + \sqrt {4 + \cdots} } } }$ [equals $3$].

Notation for Pi

$3 \cdotp 14159 \, 26535 \, 89793 \, 23846 \, 26433 \, 83279 \, 50288 \, 41971 \ldots$:

Euler, who first used the Greek letter $\pi$ in its modern sense, ...

Pi: Modern Developments

$3 \cdotp 14159 \, 26535 \, 89793 \, 23846 \, 26433 \, 83279 \, 50288 \, 41971 \ldots$:

... in $1983$ a Japanese team of Yoshiaki Tamura and Tasumasa Kanada produced $16,777,216$ ($= 2^{24}$) places.

Pythagorean Triangle with Sides in Arithmetic Sequence

$5$:

The $3-4-5$ triangle is the only Pythagorean triangle whose sides are in arithmetical sequence.

Sam Loyd's Missing Square

$5$:

Simson also discovered the identity $F_{n - 1} F_{n + 1} - {F_n}^2 = \paren {-1}^n$, which is the basis of a puzzling trick first presented by Sam Loyd.

Fibonacci Number as Sum of Binomial Coefficients

$5$:

Lucas discovered a relationship between Fibonacci numbers and the binomial coefficients:

$F_{n + 1} = \dbinom n 0 + \dbinom {n - 1} 1 + \dbinom {n - 2} 1 + \cdots$

Corollary to Euler-Binet Formula

$5$:

There is another version of this formula, which is simpler to use in practice. Because

$\dfrac 1 {\sqrt 5} \paren {\dfrac {1 + \sqrt 5} 2}^n$

is only $0 \cdotp 618 \ldots$ when $n = 1$ and rapidly becomes very small indeed, $F_n$ is actually the nearest integer to

$\paren {\dfrac {\sqrt 5 - 1} 2}^n$

No $4$ Fibonacci Numbers can be in Arithmetic Sequence

$5$:

(Incidentally, no four terms of the Fibonacci sequence can be in arithmetic progression.)

Number of Fibonacci Numbers with Same Number of Decimal Digits

$5$:

The number of Fibonacci numbers having the same number of digits is either $4$ or $5$.

Sequence of Fibonacci Numbers ending in Index

$5$:

The sequence of numbers, $n$, such that $F_n$ ends in $n$, starts: $1 \ 5 \ 25 \ 29 \ 41 \ 49 \ldots$

Perfect Number is Sum of Successive Odd Cubes except $6$

$6$:

[ $6$ ] is the only perfect number that is not the sum of successive cubes.

Set of $3$ Integers each Divisor of Sum of Other Two

$6$:

$1$, $2$, $3$ is also the only set of $3$ integers such that each divides the sum of the other two.

Solutions to $x^3 + y^3 + z^3 = 6 x y z$

$6$:

The equation $x^3 + y^3 + z^3 = 6 x y z$ has the unique solution $x = 1$, $y = 2$, $z = 3$.

Only Number which is Sum of $3$ Factors is $6$

$6$:

It is the only number that is the sum of exactly $3$ of its factors, ...

Divisibility of Elements of Pythagorean Triple by $7$

$7$:

If $a$, $b$ are the shorter sides of a Pythagorean triangle, then $7$ divides one of $a$, $b$, $a - b$ or $a - b$.

Historical Note on the St. Ives Problem

$7$:

Pierce comments that it seems to be of the same origin as the House that Jack Built, and that Leonardo uses the same numbers as Ahmes and makes his calculations in the same way.

Definition of Deltahedron

$8$:

A deltahedron is a polyhedron all of whose faces are triangular.

Relation between Square of Fibonacci Number and Square of Lucas Number

$11$:

Squaring the Fibonacci numbers, then alternately subtracting and adding $4$, produces the squares of the Lucas numbers:

$5 \times 1^2 - 4 = 1^2 \qquad 5 \times 1^2 + 4 = 3^2$

$5 \times 2^2 - 4 = 2^2 \qquad 5 \times 3^2 + 4 = 7^2 \qquad$ and so on.

Square of Hypotenuse of Pythagorean Triangle is Difference of two Cubes

$13$:

The square of the hypotenuse of a right-angled triangle is also the difference of $2$ cubes; thus, $13^2 = 8^3 - 7^3$.

Solutions of Ramanujan-Nagell Equation

$15$:

The equation $x^2 + 7 = 2^n$ has solutions for only $5$ values of $n$: $3, 4, 5, 7$ and $15$.

Product of Two Triangular Numbers to make Square

$15$:

For every triangular number, $T_n$, there are an infinite number of other triangular numbers, $T_m$, such that $T_n T_m$ is a square. For example, $T_3 \times T_{24} = 30^2$.

Triangular Number Pairs with Triangular Sum and Difference

$15$:

$15$ and $21$ are the smallest pair of triangular numbers whose sum and difference ($6$ and $36$) are also triangular. The next such pairs are $780$ and $990$, and $1,747,515$ and $2,185,095$.

Integers with Prime Values of Divisor Sum

$16$:

$\map \sigma {16} = 31$ is prime. The sequence of $n$ for which $\map \sigma n$ is prime runs: $2, 4, 9, 16, 25, 289 \ldots$

Historical Note on Hexadecimal Notation

$16$:

J.W. Mystrom in the nineteenth century proposed that the numbers $1$ to $16$ in this system ...

Smallest Odd Number not of form $2 a^2 + p$

$17$:

[$17$ is] the smallest odd number which cannot be represented as the sum of a prime and twice a square.

Stronger Feit-Thompson Conjecture

$17$:

The only known prime values for which $p^p - 1$ and $q^q - 1$ have a common factor less than $400,000$ are $17$ and $3313$. The common factor is $112,643$.

Only Number Twice Sum of Digits is $18$

$18$:

$18$ is the smallest number that is twice the sum of its digits.

Prime to Own Power minus $1$ over Prime minus $1$ being Prime

$19$:

$\paren {19^{19} - 1} / \paren {19 - 1}$ is prime. Other primes with this property are $2$, $3$, $7$ and $31$.

Semiperfect Number

$20$:

$20$ is the $2$nd semi-perfect number or pseudonymously pseudoperfect number, because it is the sum of some of its own factors: $20 = 10 + 5 + 4 + 1$.

$23$ is Largest Integer not Sum of Distinct Powers

$23$:

$23$ is the largest integer that is not the sum of distinct powers.

Smallest Integer not Sum of Two Ulam Numbers

$23$:

[$23$ is] the smallest number which is not the sum of two Ulam numbers.

Numbers with Square-Free Binomial Coefficients

$23$:

For every $n$ greater than $23$, none of the binomial coefficients $\dbinom n k$ are square-free.

Apothecaries' Ounce

$24$:

... Also $24$ scruples in an ounce, and ...

$24$ is Smallest Composite Number the Product of whose Proper Divisors is Cube

$24$:

The smallest composite number, the product of whose proper divisors is a cube. $2 \times 3 \times 4 \times 6 \times 8 \times 12 = 24^3$.

Sociable Chain: $12,496$

$28$:

The longest known sociable chain is of $28$ links, starting with $12,496$.

Historical Note on Definition of Perfect Number: Mistake $1$

$28$:

The first $4$ perfect numbers, $6$, $28$, $496$ and $8218$, were known to the late Greeks.

Historical Note on Definition of Perfect Number: Mistake $2$

$28$:

... Iamblichus, not unnaturally bearing in mind that he had no conception of the number base $10$ as mathematically arbitrary, conjectured that there was one perfect number for each number of digits, and further that they not only ended in either $6$ or $8$, which is true, but that the $6$s and $8$s alternate, which is not.

Sequence of Prime Primorial minus $1$

$29$:

Primorial $(n) - 1$ is prime for $3$, $5$, $11$, $13$, $41$, $89$, $317$, $991$, $1873$, $2053$, and no other values below $2377$.

Schatunowsky's Theorem

$30$:

$30$ is the greatest number such that all the numbers less than it and prime to it are themselves primes. The other numbers with this property are $2$, $3$, $4$, $6$, $8$, $12$, $18$ and $24$.

Dodecahedron is Dual of Icosahedron

$30$:

The dodecahedron and its dual, the icosahedron, each have $30$ edges.

Smallest Positive Integer not of form $\pm 4 \pmod 9$ not representable as Sum of Three Cubes

$30$:

$30$ is the smallest number which has not been represented as the sum of $3$ integer cubes.

Giuga Number

$30$:

$1/2 + 1/3 + 1/5 - 1/30 = 1$, so $30$ is a Guiga number, the smallest.

Smallest Set of Weights for Two-Pan Balance

$31$:

Using both pans, the solution is similar, but now relies on expressing the weight as the sum and difference of powers of $3$. With the weights $1$, $3$, $9$ and $27$ it is possible to weight up to $40$. In general the weights up to $3$ will weigh up to a maximum of $\frac 1 2 \paren {3^{n + 1} - 1}$.

Integer as Sum of $5$ Non-Zero Squares

$33$:

Any integer greater than $33$ can be written as the sum of $5$ non-zero squares. [ Jackson, Masat, Mitchell, MM v61 41 ]

Triplets of Products of Two Distinct Primes

$33$:

The triplet $33, 34, 35$ is the smallest in which each number is the product of $2$ distinct primes. ... The next such triplets are: $93, 94, 95$; $141, 142, 143$; $201, 202, 203$; $213, 214, 215$; $217, 218, 219$; ...

Prime Factors of $35$, $36$, $4734$ and $4735$

$35$:

$35$ and $4375$ have the same prime factors between them (namely $2$, $3$, $5$ and $7$) as have $36$ and $4374$.

Element of Pascal's Triangle is Sum of Diagonal or Column starting above it going Upwards

$35$:

In Pascal's Triangle, each number is the sum of either of the diagonals starting immediately above it, and taking the long way to the edge: for example, $35 = 15 + 10 + 6 + 3 + 1$.

Square Numbers which are Divisor Sum values

$36$:

The sequence of squares which are values of $\map \sigma n$ goes: $1 \ \ 4 \ \ 36 \ \ 121 \ \ 144 \ \ 256 \ \ 576 \ \ldots$

Hilbert-Waring Theorem: $5$

$37$:

Every number is the sum of at most $37$ $5$th powers.

Euler Lucky Number: $41$

$41$:

[ $x^2 + x + 41$ ] has many further prime values, including $581$, among its first $1000$ values.

Non-Palindromes in Base $2$ by Reverse-and-Add Process

$43$:

In base $2$, $43 = 101011$. This base $2$ number never becomes a palindrome by the reverse-and-add process.

Subfactorial: $5$

$44$:

Subfactorial $5 = 5! \paren {1 - 1 / {1!} + 1 / {2!} - 1 / {3!} - 1 / {4!} - 1 / {5!} } = 44$

Sequence of Kaprekar Triples

$45$:

The sequence of such triples starts: $1 \quad 8 \quad 10 \quad 45 \quad 297 \quad 2322 \ldots$

$46$: Historical Note

$46$:

... in Psalm $46$, the $46$th word is 'shake'. The $46$th word from the end counting backwards is 'spear'. Shakespear!

Prime between $n$ and $9 n$ divided by $8$

$48$:

If $n$ is greater than $48$, then there is a prime between $n$ and $9 n / 8$, inclusive.

Hilbert-Waring Theorem: $4$

$53$:

Every positive integer is the sum of at most $53$ $4$th powers.

Primes not Sum of or Difference between Powers of $2$ and $3$

$53$:

[$53$ is] The smallest prime which is not the difference between powers of $2$ and $3$. The sequence of such primes continues: $71 \quad 103 \quad 107 \quad 109 \quad 149 \quad 151 \ldots$

Even Integer with Abundancy Index greater than $9$

$55$:

Every even number $n$ whose abundancy index, defined to equal $\map \sigma n / n$, is greater than $9$, has at least $55$ distinct prime factors.

Definition:Highly Composite Number

$60$:

[$60$ is] the $8$th 'highly composite' number, defined by Ramanujan as a number that, counting from $1$, sets a record for the number of its divisors ... The sequence of 'highly composite' numbers starts: $2 \quad 4 \quad 6 \quad 12 \quad 24 \ldots$

Kaprekar's Process for $2$-Digit Numbers

$63$:

Kaprekar's process for $2$-digit numbers leads to the cycle $63 - 27 - 45 - 9 - 81 \ldots$

Existence of Number to Power of Prime Minus $1$ less $1$ divisible by Prime Squared

$64$:

For every prime $p$, there are values of $a$ such that $a^{p - 1} = 1$ is actually divisible by ${p^2}$. The smallest such value for $p = 3$ is $8^2 = 64$: $64 - 1$ is divisible by $3^2 = 9$.

Prime Numbers which Divide Sum of All Lesser Primes

$71$:

The numbers $2$, $5$, $71$, $369,119$ and $415,074,643$ are the only known numbers that divide the sum of all the primes less than them.

$4$ Positive Integers in Arithmetic Sequence which have Same Euler Phi Value

$72$:

$\map \phi {72} = \map \phi {78} = \map \phi {84} = \map \phi {90} = 24$. This is the smallest set of four numbers in arithmetical progression whose $\phi$ values are equal. The next two $4$-term arithmetical progressions with equal $\phi$ values start at $216$ and $76,236$ and each has a common difference, $6$.

Smallest $5$th Power equal to Sum of $5$ other $5$th Powers

$72$:

$72^5 = 19^5 + 43^5 + 46^5 + 47^5 + 67^5$ is the smallest $5$th power equal to the sum of $5$ other $5$th powers.

Numbers equal to Sum of Primes not Greater than its Prime Counting Function Value

$100$:

The largest number for which the sum of the primes less than the number of primes less than or equal to the number is the number itself. In this case, $\map \pi n = 25$, and the sum of the primes from $2$ to $23 = 100$. The other numbers with this property are $5$, $17$, $41$ and $77$.

Integers such that Difference with Power of $2$ is always Prime

$105$:

Erdős conjectured that [105] is the largest number $n$ such that the positive values of $n - 2^k$ are all prime. The only other known numbers with this property are $7$, $15$, $21$, $45$ and $75$.

Odd Integers whose Smaller Odd Coprimes are Prime

$105$:

$105$ is the largest integer such that every odd integer less than it and prime to it is a prime number.

Integers whose Divisor Sum equals Half Phi times Divisor Count

$105$:

$105$ is the second number $n$ such that $\map \phi n \times \map \nu n = \map \sigma n$, where $\map \nu n$ is the number of divisors of $n$. $\map \phi {105} = 48$, $\map \nu {105} = 8$ and $\map \sigma {105} = 192$.

The first such number is $35$.

Reciprocals of Odd Numbers adding to $1$

$105$:

There are $4$ ways of representing $1$ as the sum of odd reciprocals, using only $9$ of them ...

Divisor Count of $108$

$108$:

The number of its factors is also a cube: $8$.

Integers whose Divisor Sum is Cube

$110$:

$\map \sigma {110} = 216 = 6^3$, the first $n$, after $n = 1, 7$, for which $\map \sigma n$ is a cube.

Difference between Two Squares equal to Repunit

$111$:

$111 = 20^2 - 17^2$, the third difference of $2$ squares equal to a repunit. The sequence of such squares starts $1, 0$; $6, 5$; $20, 17$; $56, 45$; $156, 115$; $344, 85$; $356, 125 \ldots$

Smallest Number which is Sum of $4$ Triples with Equal Products

$118$:

$118$ is the smallest number which can be written as the sum of four triples, whose products are all equal: $118 = 14 + 50 + 54 = 15 + 40 + 63 = 18 + 30 + 70 = 21 + 25 = 72$.

The product of each triple is $37,800$. [Mauldron, Guy, 172]

Sum of Cubes of $5$ Consecutive Integers which is Square

$118$:

The sum of the cubes of the $5$ consecutive positive integers $118$ to $122$ is a square.

The only other similar sequences with this property start with $1$ and $96$.

Triperfect Number

$120$:

Only $6$ tri-perfect numbers are known:

$120, \quad 672, \quad 523,776, \quad 459,818,240, \quad 1,476,304,896, \quad 31,001,180,160$

Multiply Perfect Number of Order $8$

$120$:

One of the smallest [multiply perfect numbers] of order $8$ was discovered by Alan L. Brown, an American 'human computer': $2 \times 3^{23} \times 5^9 \times 7^{12} \times 11^3 \times 13^3 \times 17^2 \times$

$19^2 \times 23 \times 29^2 \times 31^2 \times 37 \times 41 \times 53 \times 61 \times 67^2 \times 71^2 \times 73 \times 83$

$\times 89 \times 103 \times 127 \times 131 \times 149 \times 211 \times 307 \times 331 \times 463 \times 521$

$\times 683 \times 709 \times 1279 \times 2141 \times 2557 \times 5113 \times 6481 \times 10,429$

$\times 20,857 \times 110,563 \times 599,479 \times 16,148,168,401$.

Square Numbers which are Sum of Consecutive Powers

$121$:

$121$ is the only square number that is the sum of consecutive powers from $1$: $121 = 1 + 3 + 9 + 27 + 81$.

Numbers whose Difference equals Difference between Cube and Seventh Power

$125$:

$5^3 - 2^7 = 5 - 2$. The only known matching pattern is $13^3 - 3^7 = 13 - 3$.

Triangles with Integer Area and Integer Sides in Arithmetical Sequence

$126$:

[$126$ is] the area of the third-smallest triangle with integral sides in arithmetical sequence and integral area: the sides are $15, 28, 41$. The first two are $3, 4, 5$ and $13, 14, 15$; the next is $15, 26, 37$, with area $156$.

Sequence of Quasiamicable Pairs

$140$:

With $195$, the second pair of betrothed or quasi-amicable numbers. The first pair is $\tuple {48, 95}$, the third $\tuple {1575, 1648}$.

Carmichael's Theorem

$144$:

A divisor of a Fibonacci number is called proper if it does not divide any smaller Fibonacci number. The only Fibonacci numbers that do not possess a proper divisor are $1$, $8$ and $144$.

Smallest Prime Magic Square with Consecutive Primes from $3$

$144$:

The smallest magic square composed of consecutive primes comprises the $144$ odd primes from $3$ upwards. The magic constant is $4515$.

Sum of $2$ Squares in $2$ Distinct Ways: $145$

$145$:

The $5$th number to be the sum of $2$ squares in $2$ different ways:

$145 = 12^2 + 1^2 = 8^2 + 9^2$

Sequence of Square Centered Hexagonal Numbers

$169$:

The smallest square hexagonal number, apart from $1$ [is $169$]. The next smallest are $32, 761$ and $6,355,441$.

$169$ as Sum of up to $155$ Squares

$169$:

In fact, $169$ can be written as the sum of $n$ non-zero squares, for all values of $n$ from $1$ to $155$, but for no larger values. [ Jackson, Masat and Mitchell, MM v61 41 ]

$3$-Digit Numbers forming Longest Reverse-and-Add Sequence

$187$:

The smallest of a group of $3$-digit numbers that require $23$ reversals to form a palindrome.

Numbers such that Divisor Count divides Phi divides Divisor Sum

$210$:

$\map \phi {210} = 48$ is a factor of $\map \sigma {210} = 576$, and $\map d {210} = 16$ divides both. The sequence of numbers with both these properties starts: $1 \quad 3 \quad 15 \quad 30 \quad 35 \quad 52 \quad 70 \quad 78 \quad 105 \quad 140 \quad 168 \quad 190 \quad 210 \ldots$

Smallest Order $3$ Multiplicative Magic Square: Historical Note

$216$:

$216$ is the magic constant in the smallest possible multiplicative magic square, as discovered by Dudeney.

Plato's Geometrical Number

$216$:

The famous and notorious number of Plato occurs in an obscure passage in The Republic, $\text{viii}$, $546$, $\text {B - D}$ ... Adams eventually reaches the conclusion that the number intended in the quoted passage is $216$ as the sum of the cubes of the sides of the triangle ... [J. Adams, The Republic of Plato, CUP, 1929]

Amicable Pairs with Common Factor $3$

$220$:

Most known amicable pairs have both numbers in the pair divisible by $3$.

Amicable Pair with Smallest Common Prime Factor $5$

$220$:

Most known amicable pairs have both numbers in the pair divisible by $3$. However, this is not a general rule: this counterexample by te Riele may be the smallest such: $5 \times 7^2 \times 11^2 \times 13 \times 17 \times 19^3 \times 23 \times 37 \times 181$ multiplied by either $101 \times 8643 \times 1 \, 947 \, 938 \, 229$ or by $365 \, 147 \times 47 \, 303 \, 071 \, 129$.

Solution of Ljunggren Equation

$239$:

The only solution of the equation $x^2 + 1 = 2 y^2$ is $x = 239, y = 13$.

Solutions of Diophantine Equation $x^4 + y^4 = z^2 + 1$ for $x = 239$

$239$:

The 'approximation' to a Fermat equation, $x^4 + y^4 = z^4 + 1$, has $3$ solutions with $x = 239$. The other numbers are $y = 104, z = 58, 136$; $y = 143, z = 60,671$; $y = 208, z = 71, 656$.

Prime Decomposition of $7$th Fermat Number

$257$:

Thus, in $1909$, Moorhead and Western proved that $F_7$ and $F_8$ are composite, without producing any factors.

Pépin's Test

$257$:

Such tests are easily performed today on computers using this criterion, which is similar to Lucas's test for the primality of Mersenne numbers: $F_n$ is prime if and only if it divides $3^{1/x \paren {F_n - 1} } + 1$.

Consecutive Powerful Numbers

$288$:

With $289$, the second pair of consecutive powerful numbers: $288 = 2^5 3^2$ and $289 = 17^2$. The smallest such pair is $8, 9$ and the next two pairs are $9800, 9801$ and $332, 928, \ 332, 929$.

297

$297$:

[$297$ is] the $5$th Kaprekar number.

$1,111,111,111$

$297$:

So is $1,111,111,111$ [a Kaprekar number ], the smallest Kaprekar number of $10$ digits whose square is $12, 345, 671, 900, 987, 654, 321$.

Fourth Powers which are Sum of $4$ Fourth Powers

$353$:

$353^4$ is the smallest number that is the sum of $4$ other $4$th powers ... The sequence of such numbers continues: $651 \quad 2487 \quad 2501 \quad 2829 \ldots$

Lucas-Carmichael Number

$399$:

The smallest Lucas-Carmichael number $n$, such that if $p$ divides $n$, then $p + 1$ divides $n + 1$.

Largest Number not Expressible as Sum of Less than $37$ Positive Fifth Powers

$466$:

The largest number which cannot be represented with less than $32$ $5$th powers.

Solutions to $p^2$ Divides $10^p - 10$

$487$:

One of the only $3$ primes, less than $2^{32}$ such that $p^2$ divides $10 p - 10$. The others are $3$ and $56,598,313$.

Kaprekar's Process on $3$ Digit Number ends in $495$

$495$:

Take any $3$-digit number whose digits are not all the same and is not a palindrome. Arrange its digits into ascending and descending order and subtract. Repeat. This is called Kaprekar's process. All $3$-digit numbers eventually end up with $495$.

Products of $2$-Digit Pairs which Reversed reveal Same Product

$504$:

$504$ is equal to both $12 \times 42$ and $21 \times 24$. There are thirteen such $2$-digit pairs, the largest being $36 \times 84 = 63 \times 48 = 3024$.

Prime Decomposition of $5$th Fermat Number

$641$:

Euler found the first counterexample to Fermat's conjecture that $2^{2^n} + 1$ is always prime, when he discovered in $1742$ that $2^{2^5} + 1$ is divisible by $641$.

Tetrahedral Numbers which are Sum of $2$ Tetrahedral Numbers

$680$:

$680$ is the smallest tetrahedral number to be the sum of two tetrahedral numbers: $680 = 120 + 560$.

Consecutive Integers whose Product is Primorial

$714$:

They discovered on computer that only primorial $1$, $2$, $3$, $5$ and $7$ can be represented as the product of consecutive numbers, up to primorial $3049$.

$720$ is Product of Consecutive Numbers in Two Ways

$720$:

$10! = 7! \times 6!$, the only known example of a factorial being the product of two other factorials.

Solutions to Approximate Fermat Equation $x^3 = y^3 + z^3 \pm 1$

$729$:

Mistake $1$:

$9^3$ and the $2$nd smallest cube to be the sum of $3$ cubes: $9^3 = 1^3 + 6^3 + 8^3$. This makes $7^3$ the smallest solution to the approximate Fermat equation, $x^3 = y^3 + z^3 + 1$.

Mistake $2$:

The next such solution is $104^3 = 64^3 + 94^3 + 1$.

Period of Reciprocal of $729$ is $81$

$729$:

$1 / 729$ has a decimal period of $81$ digits, which can be arranged in groups of $9$ digits, reading across each row, in this pattern:

$\ds 001 \, 371 \, 742$

$\ds $

$\ds 112 \, 482 \, 853$

$\ds $

$\ds 223 \, 593 \, 964$

$\ds $

$\ds 334 \, 705 \, 075$

$\ds $

$\ds 445 \, 816 \, 186$

$\ds $

$\ds 556 \, 927 \, 297$

$\ds $

$\ds 668 \, \color {red} 6 38 \, 408$

$\ds $

$\ds 779 \, 149 \, 519$

$\ds $

$\ds 890 \, 260 \, 631$

$\ds $

Sum of 4 Consecutive Binomial Coefficients forming Square

$767$:

Mistake $1$:

$\dbinom {767} 1 + \dbinom {767} 2 + \dbinom {767} 3 + \dbinom {767} 4$ is a perfect square, $8672^2$.

Mistake $2$:

The smaller solutions are $7$, $15$ and $74$.

Triangular Number Pairs with Triangular Sum and Difference: $T_{39}$ and $T_{44}$

$780$:

$780$ and $990$ are the $2$nd smallest pair of triangular numbers whose sum and difference ($1770$ and $210$) are also triangular.

Multiple of $999$ can be Split into Groups of $3$ Digits which Add to $999$

$999$:

In fact, any multiple at all of $999$ can be separated into groups of $3$ digits from the unit position, which when added will total $999$. The same principle applies to multiples of $9 \quad 99 \quad 9999$ and so on.

Numbers Not Expressible as Sum of no more than $5$ Composite Numbers

$1167$:

[$1167$ is] The largest of $256$ numbers which cannot be expressed as the sum of at most $5$ composite numbers. [Guy 136]

Integer both Square and Triangular

$1225$:

It is the second number to be simultaneously square and triangular.

Square Numbers which are Sum of Sequence of Odd Cubes

$1225$:

$35^2 = 1^3 + 3^3 + 5^3 + 7^3 + 9^3$. The next such sum is $1^3 + \ldots 29^3$.

1477

$1477$:

$1477! + 1$ is prime. $n! + 1$ is also prime for $1$, $2$, $3$, $11$, $27$, $37$, $41$, $73$, $77$, $116$, $154$, $320$, $340$, $399$, $427$, $872$, $1477 \ldots$

$17$ Consecutive Integers each with Common Factor with Product of other $16$

$2185$:

The start of a sequence of $17$ consecutive integers, each of which has a common factor, greater than $1$, with the product of the remaining $16$.

$2601$ as Sum of $3$ Squares in $12$ Different Ways

$2601$:

This square can be represented as the sum of $3$ squares in $10$ distinct ways. The largest smallest-square solution is $24^2 + 27^2 + 36^2$. The smallest largest-square solutions are $34^2 + 31^2 + 22^2$ and $34^2 + 34^2 + 17^2$.

3000

$3000$:

The smallest number requiring more than $12$ letters to write in the English language. It uses $13$.

Perfect Digit-to-Digit Invariant: $3435$

$3435$:

$3435 = 3^3 + 4^4 + 3^3 + 5^5$. This is the only known number with this property, apart from $1^1 = 1$.

Product with Repdigit can be Split into Parts which Add to Repdigit

$6666$:

... if a number is multiplied by a number whose digits are all the same, for example, let $894$ be multiplied by $22,222$, then in this case the right-hand $5$ digits, added to the left-hand portion, form another number with equal digits: $894 \times 22,222 = 19866468$ and $198 + 66,468 = 66666$.

$6667$

$6667$:

Not a mistake as such, but:

The patterns appearing in $6667^2$, and similarly in $3334^2$ and so on, are examples of a general rule. Any number, of however many digits, will form a pattern when a sufficiently large number of either $3$s, $6$s or $9$s are prefixed to it. Thus, $72^2 = 5184$, $672^2 = 451, 584$ and $6672^2 = 44, 515, 584$ and so on.

is so vaguely worded as to be all but useless.

$8712$

$8712$:

It is a multiple of its reversal, $2718$.

Number of Different Ways to play First $n$ Moves in Chess

$8902$:

There are $8902$ ways of playing the first $4$ moves at chess.

$9801$

$9801$:

$9801 = 99^2$ and $98 + 01 = 99$, so $9801$ is a Kaprekar number.

Smallest Penholodigital Square

$11,826$:

$11,826^2$ is the smallest pandigital square. It was first noted by John Hill in $1727$, who thought it was the only pandigital square.

Smallest Integer which is Product of $4$ Triples all with Same Sum

$25,200$:

The smallest number which can be written as the product of $4$ triples, each with the same sum ... The common sum is $137$. [Mauldron, AMM v88]

Smallest Fermat Pseudoprime to Bases $2$, $3$, $5$ and $7$

$29,351$:

[$29,351$ is] Simultaneously a pseudoprime in bases $2$, $3$, $5$ and $7$.

Pentagonal and Hexagonal Numbers

$40,755$:

The first number, after $1$ and $15$, to be simultaneously pentagonal and hexagonal and therefore, also, triangular.

Carmichael Number with $4$ Prime Factors

$41,041$:

The smallest Carmichael number with $4$ factors. It equals $7 \times 11 \times 13 \times 41$.

Smallest Fourth Power as Sum of $5$ Distinct Fourth Powers

$50,625$:

Equal to $15^4 = 4^4 + 6^4 + 8^4 + 9^4 + 14^4$.

This is the smallest example of a $4$th power equal to the sum of only $5$ other $4$th powers.

Ackermann Function: $1$

$65,536$:

The Ackermann function is one of the fastest increasing functions used in mathematics. Its values from $\map f 0$ to $\map f 5$ are $1$, $3$, $4$, $8$, $65,536$.

Property of $74,162$

$74,162$:

$74,162^2 = 5,500,002,244$ [JRM v17 84]

Sets of $4$ Prime Quadruples

$99,131$:

There are $35$ sets of $4$ consecutive prime numbers of the form $10 n + 1$, $10 n + 3$, $10 n + 7$, $10 n + 9$, below $100,000$.

Kaprekar's Process on $5$ Digit Number

$99,954$:

Kaprekar's process for all $5$-digit numbers whose digits are not all equal leads to one of $3$ separate cycles. The smallest cycle is $99,954 - 95,553$. The other two cycles are $98,532 - 97,443 - 96,642 - 97,731$ and $98,622 - 97,533 - 96,543 - 97,641$.

Numbers whose Cube equals Sum of Sequence of that many Squares

$103,823$:

$103,823 = 47^3 = 22^2 + 23^2 + \ldots + 67^2 + 68^2$, the smallest representation of a cube as the sum of consecutive squares. The next smallest is $2161^3$.

Number times Recurring Part of Reciprocal gives $9$-Repdigit

$142,857$:

This is a property of all the periods of repeating decimals. If the period of $n$ is multiplied by $n$, the result is as many $9$s as there are digits in the period of $1/n$.

Reciprocal of $142 \, 857$

$142,857$:

Because $142,857 \times 7 = 999,999$, the decimal period of $1 / 7$ is $142857$ and the decimal period of $1 / 142,857$ is $7$. In fact $1 / 142,857 = 0.000007 \, 000007 \, 000007 \, \ldots$

Integer whose Digits when Grouped in $3$s add to Multiple of $999$ is Divisible by $999$

$142,857$:

Now any number whose digits when grouped in $3$s from the units end add up to $999$ is a multiple of $999$, and conversely, so $142857$ must be a multiple of $999$.

$147 \, 852$

$147,852$:

The digits $147852$ in various orders that are not permutations of the period of $1/7$ occur in several other products also.

$517 \, 842$

$147,852$:

The digits $147852$ in various orders ... occur in several other products also. For example, $666 \times 777 = 517,842$ and $333 \times 777 = 258,741$.

Properties of Family of $333,667$ and Related Numbers

$333,667$:

The same author shows other patterns involving the same number: ...

$3,333,366,667 \times 1,111,333 = 371,113,711,137,111$ and so on.

Definition:Rare Number

$621,770$:

$621,770 + 621,770 = 836^2$and $621,770 - 077,126$ (its reversal) $= 738^2$. The only other number with this property, less than $10^8$ is $65$: $65 + 65 = 11^2$ and $65 - 56$ (its reversal) $= 3^2$.

Triangular Number Pairs with Triangular Sum and Difference: $T_{1869}$ and $T_{2090}$

$1,747,515$:

Together with $2,185,095$ the $3$rd pair of triangular numbers whose sum and difference are also triangular.

Factorial as Product of Consecutive Factorials

$3,628,800$:

... the only factorial that is the product of other consecutive factorials apart from the trivial $1! = 0! \times 1!$, $2! = 0! \times 1! \times 2!$ and $1! \times 2! = 2!$.

Archimedes' Cattle Problem

$4,729,494$:

... in this case the total number of cattle is a number of $206,545$ digits, starting $7766 \ldots$

Sequence of Triplets of Primitive Pythagorean Triangles with Same Area

$13,123,110$:

The area of the next such triplet is $2,570,042,985,510$.

Hardy-Ramanujan Number: $87 \, 539 \, 319$

$87,539,319$:

The smallest number that can be represented as the sum of $2$ cubes in $3$ different ways.

Pandigital Integers remaining Pandigital on Multiplication

$123,456,789$:

There are several numbers that are pandigital, including zero, and remain so when multiplied by several factors. For example, $1,098,765,432$ when multiplied by $2$, $4$, $5$ or $7$.

Triangular Numbers which are Product of $3$ Consecutive Integers

$258,474,216$:

The largest triangular number to be the product of consecutive integers. The others are $6$, $120$, $210$, $990$ and $185 \, 136$.

Polydivisible Number: $381 \, 654 \, 729$

$381,654,729$:

The unique integer such that the number formed by the first $n$ digits is divisible by the digit $n$.

Palindromic Smith Number: $123,455,554,321$

$12,345,554,321$:

[$12,345,554,321$ is] a palindromic Smith number.

General Fibonacci Sequence whose Terms are all Composite: $1$

$62,638,280,004,239,857$:

The first term of a generalized Fibonacci sequence, in which each term is the sum of the previous two, in which every term is composite. The second term is $49,463,435,743,205,655$.

Square of Small Repunit is Palindromic

$1,111,111,111,111,111,111$:

The squares of repunits make a pretty pattern:

For example: ... $1,111,111,111^2 = 12,345,678,900,987,654,321$

Smallest Number which is Multiplied by $99$ by Appending $1$ to Each End

$112,359,550,561,797,732,809$:

The smallest number which, when $1$ is placed at both ends, the number is multiplied by $99$.

Sequence of $9$ Consecutive Integers each with $48$ Divisors

$17,796,126,877,482,329,126,044$:

The first in a sequence of $9$ consecutive integers, each with $9$ divisors.

General Fibonacci Sequence whose Terms are all Composite: $2$

$1,786,772,701,928,802,632,268,715,130,455,793$:

Together with $1,059,683,225,053,915,111,058,165,141,686,996$, the start of a generalized Fibonacci sequence (in which each term is the sum of the previous two) in which every member is composite although the first $2$ terms have no common factor.

Integer which is Multiplied by $9$ when moving Last Digit to First

$10,112,359,550,561,797,752,808,988,764,044,943,820,224,719$:

The unique number which, when the last digit, $9$, is moved to the front, is the number multiplied by $9$.

Upper Bound for Number of Grains of Sand to fill Universe

$10^{51}$:

... the number of grains of sand required to fill the universe turns out to be, in our notation, less than $10^{51}$.

Definition:Fischer-Griess Monster: Historical Note

$808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000$:

... Monster sporadic group, discovered by Fischer in $1974$.

Amicable Pair: Examples: $59 \, 554 \, 936 \ldots \, 048 \, 448 - 59 \, 554 \, 936 \ldots 105 \, 472$

$2^4 \times 7 \times 9,288,811,670,405,087 \times 145,135,534,866,431 \times 313,887,523,966,328,699,903$:

Together with its amicable friend, $2^4 \times 7 \times 9,288,811,670,405,087 \times 45,556,233,678,753,109,045,286,896,851,222,527$, the largest known pair of amicable numbers. [ Yan and Jackson, Computer Mathematics and Applications v27]

Factorisation of $\paren {11^{104} + 1} / \paren {11^8 + 1}$

$\paren {11^{104} + 1} / \paren {11^8 + 1}$:

Equal to $86,759,222,313,428,390,812,218,077,095,850,708,048,977$ $\times$ $1,084,881,048,536,374,706,129,613,998,429,729,484,098,346,115,257,905,772,116,753$.

Mersenne Prime $M_{521}$

$2^{521} - 1$:

In a few hours on the night of $30$ January $1952$, using the SWAC computer, Lehmer proved that $2^{521} - 1$ and the $183$-digit number $2^{607} - 1$ are both Mersenne primes.