Diophantus of Alexandria/Arithmetica

Book $\text I$

Problem $1$

To divide a given number into two having a given difference.

Problem $2$

Diophantus of Alexandria/Arithmetica/Book 1/Problem 2

Problem $3$

Diophantus of Alexandria/Arithmetica/Book 1/Problem 3

Problem $4$

Diophantus of Alexandria/Arithmetica/Book 1/Problem 4

Problem $5$

Diophantus of Alexandria/Arithmetica/Book 1/Problem 5

Problem $6$

Diophantus of Alexandria/Arithmetica/Book 1/Problem 6

Problem $7$

Diophantus of Alexandria/Arithmetica/Book 1/Problem 7

Problem $8$

What number must be added to $100$ and to $20$ (the same number added to each) so that the sums are in the ratio $3 : 1$?

Problem $9$

Diophantus of Alexandria/Arithmetica/Book 1/Problem 9

Problem $10$

Diophantus of Alexandria/Arithmetica/Book 1/Problem 10

Problem $11$

Diophantus of Alexandria/Arithmetica/Book 1/Problem 11

Problem $12$

Diophantus of Alexandria/Arithmetica/Book 1/Problem 12

Problem $13$

Diophantus of Alexandria/Arithmetica/Book 1/Problem 13

Problem $14$

Diophantus of Alexandria/Arithmetica/Book 1/Problem 14

Problem $15$

Two numbers are such that:

if the first receives $30$ from the second, they are in the ratio $2 : 1$

if the second receives $50$ from the first, they are in the ratio $1 : 3$.

What are these numbers?

Problem $16$

Diophantus of Alexandria/Arithmetica/Book 1/Problem 16

Problem $17$

The sums of $4$ numbers, omitting each of the numbers in turn, are $22$, $24$, $27$ and $20$ respectively

What are the numbers?

Problem $18$

Diophantus of Alexandria/Arithmetica/Book 1/Problem 18

Problem $19$

Diophantus of Alexandria/Arithmetica/Book 1/Problem 19

Problem $20$

Diophantus of Alexandria/Arithmetica/Book 1/Problem 20

Problem $21$

Diophantus of Alexandria/Arithmetica/Book 1/Problem 21

Problem $22$

To find $3$ numbers such that, if each give to the next following a given fraction of itself, in order, the results after each has given and taken may be equal.

That is:

Let $\dfrac 1 p, \dfrac 1 q, \dfrac 1 r$ be given.

The exercise is to find a set of $3$ natural numbers $\set {x, y, z}$ such that:

\(\ds x - \frac x p + \frac z r\)

\(=\)

\(\ds m\)

\(\ds y - \frac y q + \frac x p\)

\(=\)

\(\ds m\)

\(\ds z - \frac z r + \frac y q\)

\(=\)

\(\ds m\)

where $m$ is a natural number.

Problem $23$

Diophantus of Alexandria/Arithmetica/Book 1/Problem 23

Problem $24$

Diophantus of Alexandria/Arithmetica/Book 1/Problem 24

Problem $25$

Diophantus of Alexandria/Arithmetica/Book 1/Problem 25

Problem $26$

Diophantus of Alexandria/Arithmetica/Book 1/Problem 26

Problem $27$

Diophantus of Alexandria/Arithmetica/Book 1/Problem 27

Problem $28$

Diophantus of Alexandria/Arithmetica/Book 1/Problem 28

Problem $29$

Diophantus of Alexandria/Arithmetica/Book 1/Problem 29

Problem $30$

Diophantus of Alexandria/Arithmetica/Book 1/Problem 30

Problem $31$

Diophantus of Alexandria/Arithmetica/Book 1/Problem 31

Problem $32$

Diophantus of Alexandria/Arithmetica/Book 1/Problem 32

Problem $33$

Diophantus of Alexandria/Arithmetica/Book 1/Problem 33

Problem $34$

Diophantus of Alexandria/Arithmetica/Book 1/Problem 34

Problem $35$

Diophantus of Alexandria/Arithmetica/Book 1/Problem 35

Problem $36$

Diophantus of Alexandria/Arithmetica/Book 1/Problem 36

Problem $37$

Diophantus of Alexandria/Arithmetica/Book 1/Problem 37

Problem $38$

Diophantus of Alexandria/Arithmetica/Book 1/Problem 38

Problem $39$

Diophantus of Alexandria/Arithmetica/Book 1/Problem 39

Book $\text {II}$

Diophantus of Alexandria/Arithmetica/Book 2

Book $\text {III}$

Problem $6$

To find $3$ numbers such that their sum is a square and the sum of any pair of them is a square.

That is, let $\set {p, q, r}$ be a set of $3$ natural numbers such that:

$p + q + r$ is square

$p + q$ is square

$q + r$ is square

$r + p$ is square.

What are those $3$ numbers?

Problem $12$

To find $3$ numbers such that the product of any $2$ of them added to the $3$rd gives a square.

That is, let $\set {p, q, r}$ be a set of $3$ natural numbers such that:

$p q + r$ is square

$q r + p$ is square

$r p + q$ is square

What are those $3$ numbers?

Book $\text {IV}$

Diophantus of Alexandria/Arithmetica/Book 4

Book $\text V$

Problem $30$

A man buys a certain number of measures of wine, some at $8$ drachmas, some at $5$ drachmas each.

He pays for them a square number of drachmas;

and if we add $60$ to this number, the result is a square, the side of which is equal to the whole number of measures.

Find how many he bought at each price.

Book $\text {VI}$

Diophantus of Alexandria/Arithmetica/Book 6