# Book:David Wells/Curious and Interesting Numbers

## David Wells: *The Penguin Dictionary of Curious and Interesting Numbers*

Published $\text {1986}$.

## Contents

- Introduction
- A List of Mathematicians in Chronological Sequence
- Glossary
- Bibliography

- The Dictionary

- Tables
- The First $100$ Triangular Numbers, Squares and Cubes
- The First $20$ Pentagonal, Hexagonal, Heptagonal and Octagonal Numbers
- The First $40$ Fibonacci Numbers
- The Prime Numbers less than $1000$
- The Factorials of the Numbers $1$ to $20$
- The Decimal Reciprocals of the Primes from $7$ to $97$
- The Factors of the Repunits from $11$ to $R_{40}$
- The Factors, where Composite, and the Values of the Functions $\map \phi n$, $\map d n$ and $\map \sigma n$

- Index

## Further Editions

## Errata

### 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.*

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

$2$:

*An integer is the sum of a sequence of 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 $2^1 + 2^1 = 2^2$.*

### Decimal Expansion of $\pi$

$3 \cdotp 14159 \, 26535 \, 89793 \, 23846 \, 26433 \, 83279 \, 50288 \, 41972 \ldots$:

*$3 \cdotp 14159 \, 26535 \, 89793 \, 23846 \, 26433 \, 83279 \, 50288 \, 41972 \ldots$*

### Notation for Pi

$3 \cdotp 14159 \, 26535 \, 89793 \, 23846 \, 26433 \, 83279 \, 50288 \, 41972 \ldots$:

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

### Leonhard Paul Euler

$3 \cdotp 14159 \, 26535 \, 89793 \, 23846 \, 26433 \, 83279 \, 50288 \, 41972 \ldots$:

*Euler ... gave an even more impressive demonstration of the power of these new methods by calculating $\pi$ to $20$ decimal places in just one hour.*

### Pi: Modern Developments

$3 \cdotp 14159 \, 26535 \, 89793 \, 23846 \, 26433 \, 83279 \, 50288 \, 41972 \ldots$:

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

### Tamura-Kanada Circuit Method: Example

$3 \cdotp 14159 \, 26535 \, 89793 \, 23846 \, 26433 \, 83279 \, 50288 \, 41972 \ldots$:

*Here are the values of $\pi$ after going round just $3$ times on a pocket calculator. It is already correct to $5$ decimal places!*

\(\text {(1)}: \quad\) | \(\ds \) | \(\) | \(\ds 2 \cdotp 91421 \, 35\) | |||||||||||

\(\text {(2)}: \quad\) | \(\ds \) | \(\) | \(\ds 3 \cdotp 14057 \, 97\) | |||||||||||

\(\text {(3)}: \quad\) | \(\ds \) | \(\) | \(\ds 3 \cdotp 14159 \, 28\) |

### 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:*- $F_{n + 1} = \paren {\dfrac n 0} + \paren {\dfrac {n - 1} 1} + \paren {\dfrac {n - 2} 2} + \cdots$

*For example:*

\(\ds F_{12} = 144\) | \(=\) | \(\ds \paren {\frac {11} 0} + \paren {\frac {10} 1} + \paren {\frac 9 2} + \paren {\frac 8 3} + \paren {\frac 7 4} + \paren {\frac 6 5}\) | ||||||||||||

\(\ds \qquad \ \ \) | \(\ds \) | \(=\) | \(\ds 1 + 10 + 36 + 56 + 36 + 6\) |

### No 4 Fibonacci Numbers can be in Arithmetic Sequence

$5$:

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

### 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.*

### 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, ...*

### 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.

### 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 $2,185,095$.*

### Palindromic Triangular Numbers: $1$

$15$:

*There are $40$ palindromic triangular numbers below $10^7$.*

### Stronger Feit-Thompson Conjecture

$17$:

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

### Magic Hexagon

$19$:

*There is only one way in which consecutive integers can be fitted into a magic hexagonal array ... The numbers $1$ to $19$ can be so arranged, a fact first discovered by T. Vickers.*

### Sum of Sequence of Alternating Positive and Negative Factorials being Prime

$19$:

*$19! - 18! + 17! - 16! + \dotsb + 1$ is prime. The only other numbers with this property are $3$, $4$, $5$, $6$, $7$, $8$, $10$ and $15$.*

### 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: $20 = 10 + 5 + 4 + 1$.*

### Squares Ending in 5 Occurrences of 2-Digit Pattern

$21$:

*If a square ends in the pattern $xyxyxyxyxy$, then $xy$ is either $21$, $61$ or $84$.*

### $23$ is Largest Integer not Sum of Distinct Powers

$23$:

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

### Apothecaries' Ounce

$24$:

### 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

$28$:

*... 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 $24$.*

### Pascal's Rule

$35$:

- From the rule for constructing the triangle it follows that
- $\paren {\dfrac n m} + \paren {\dfrac n {m + 1} } = \paren {\dfrac {n + 1} {m + 1} }$

### Hilbert-Waring Theorem/Particular Cases/5

$37$:

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

### 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.*

### 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 $5$, $71$ and $369119$ are the only known numbers less than $2,000,000$ that divide the sum of 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, $6$.*

### Reciprocal of 89

$89$:

*$89$ is the $11$th Fibonacci number, and the period of its reciprocal is generated by the Fibonacci sequence: $1 / 89 = 0 \cdotp 11235 \ldots$*

### 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 $75$.*

### 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 ...*

### 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':*$2 \times 3^{23} \times 5^9 \times 7^{12} \times 11^3 \times 13^3 \times 17^2 \times$- $19^2 \times 23 \times 29^2 \times 31^2 \times 37 \times 41 \times 53 \times 61 \times 67^2 \times 71^2 \times 73 \times 83$
- $\times 89 \times 103 \times 127 \times 131 \times 149 \times 211 \times 307 \times 331 \times 463 \times 521$
- $\times 683 \times 709 \times 1279 \times 2141 \times 2557 \times 5113 \times 6481 \times 10,429$
- $\times 20,857 \times 110,563 \times 599,479 \times 16,148,168,401$.

### 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 $4$th number to be the sum of $2$ squares in two different ways:*- $145 = 12^2 + 1^2 = 8^2 + 9^2$

### 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.*

### Multiplicative Magic Square/Examples/Order 3/Smallest/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 of Plato*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,*, CUP, 1929]*

### Fermat Pseudoprime to Base 4

$217$:

*The second smallest pseudoprime to base $4$ ($15$ is the smallest).**$4^{216} - 1$ is divisible by $217$ although $217$ is not prime but $7 \times 31$.*

### Prime Decomposition of 7th Fermat Number

$257$:

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

### Product of Sequence of Fermat Numbers plus 2

$257$:

*$F_{n + 1} = F_0 F_1 F_2 \ldots F_{n - 1} + 2$, from which it follows ...*

### 297

$297$:

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

### 492 is Sum of 3 Cubes in 3 Ways

$492$:

*$492$ is the sum of $3$ cubes, one or two of which may be negative, in no fewer than $10$ different ways. [Madachy]*

### 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 5th 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 $641$.*

### Consecutive Integers whose Product is Primorial

$714$:

*They discovered on computer that only primorial $1$, $2$, $3$, $4$ and $7$ can be represented as the product of consecutive numbers, up to primorial $3049$.*

### 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:*

\(\ds 001 \, 371 \, 742\) | \(\) | \(\ds \) | ||||||||||||

\(\ds 112 \, 482 \, 853\) | \(\) | \(\ds \) | ||||||||||||

\(\ds 223 \, 593 \, 964\) | \(\) | \(\ds \) | ||||||||||||

\(\ds 334 \, 705 \, 075\) | \(\) | \(\ds \) | ||||||||||||

\(\ds 445 \, 816 \, 186\) | \(\) | \(\ds \) | ||||||||||||

\(\ds 556 \, 927 \, 297\) | \(\) | \(\ds \) | ||||||||||||

\(\ds 668 \, \color {red} 6 38 \, 408\) | \(\) | \(\ds \) | ||||||||||||

\(\ds 779 \, 149 \, 519\) | \(\) | \(\ds \) | ||||||||||||

\(\ds 890 \, 260 \, 631\) | \(\) | \(\ds \) |

### 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.*

### Integer both Square and Triangular

$1225$:

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

### Squares whose Digits can be Separated into 2 other Squares

$1444$:

*$1444$ is also the $4$th square whose digits form two other squares juxtaposed, $1444 = 144 : 4$.*

### Gregorian Calendar

$3333$:

*The Gregorian calendar is approximately $1$ day ahead every $3333$ years.*

### 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.

### Mersenne Number whose Index is Mersenne Prime

$8191$:

*It had been conjectured that although most Mersenne numbers appear to be composite, a Mersenne number whose index was prime would itself be prime.*

### 9801

$9801$:

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

### Smallest Pandigital 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 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.*

### 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 $98,622 - 97,533 - 96,543 - 97,641$.*

### Recurring Part of Fraction times Period 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 $n$.*

### 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.*

### 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.*

### Palindromic Triangular Numbers: $2$

$828,828$:

*The only triangular palindrome, apart from $55$, $66$ and $666$.*

### 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 $7766 \ldots$*

### 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 $7$.*

### Right-Truncatable Prime

$739,391,133$:

*The largest prime number in base $10$ that can be 'tailed' again and again by removing its last digit to produce only primes, ending with $739$, $73$, $7$.*

### $555,555,555,555,556$

$555,555,555,555,556$:

### Square of Small Repunit is Palindromic

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

*The squares of repunits make a pretty pattern:**For example: ... $1111111111111^2 = 12345678900987654321$*

### Probability of All Players receiving Complete Suit at Bridge

$2,235,197,406,895,366,368,301,560,000$:

*The probability that all $4$ players at Bridge will be dealt a complete suit.*

### General Fibonacci Sequence whose Terms are all Composite

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

*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.*

### $180 \times \paren {2^{127} - 1} + 1$ is Prime

$180 \times \paren {2^{127} - 1} + 1$:

*The largest known prime in July $1951$, discovered by J.C.P. Miller and D.J. Wheeler of Cambridge University on the EDSAC. They had a test for numbers of the form $k \times \text M_{127} + 1$ where $\text M_{127}$ is the $127$th Mersenne number. This was the largest prime found.*

*In the same month A. Ferrier, using a desk calculator only, showed the primality of $\paren {2^{143} + 1} / 17$.*

### 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 $10^{51}$.*

### 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.*

### Ackermann Function

$2^{65536}$:

*Ackermann's function is defined by $\map f {a, b} = \map f {a - 1, \map f {a, b - 1} }$ where $\map f {1, b} = 2 b$ and $\map f {a, 1} = a$ for $a$ greater than $1$.**$\map f {3, 4} = 2^{65,536}$, which has more than $19,000$ digits.*

### Mersenne Prime $M_{86 \, 243}$

$2^{86243} - 1$:

*... the $28$th Mersenne prime, hunted down by David Slowinski on his trusty CRAY-1 in $1983$.*

### Speed of Light

$2^{86243} - 1$:

*... for example, the speed of light, $299796$ kilometres per second, is written as $10^8 \times 2.99796$ kilometres per second.*

### Horace Scudder Uhler

$9^{9^9}$:

*Horace Scuder Uhler, Professor of Physics at Yale University, devoted much of his spare time ...*

### Number of Primes up to $n$ Approximates to Eulerian Logarithmic Integral

$10^{10^{10^{34}}}$:

*The number of primes less than or equal to $n$ is approximately $\displaystyle \int_0^n \frac {\d x} {\log x}$.*