Book:Euclid/The Elements/Book IX

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Euclid: The Elements: Book IX

Published $c. 300 B.C.E$.


Contents

Book $\text{IX}$: Further Number Theory: Infinitude of Prime Numbers, Geometric Series, Perfect Numbers

Proposition $1$: Product of Similar Plane Numbers is Square
Proposition $2$: Numbers whose Product is Square are Similar Plane Numbers
Proposition $3$: Square of Cube Number is Cube
Proposition $4$: Cube Number multiplied by Cube Number is Cube
Proposition $5$: Number multiplied by Cube Number making Cube is itself Cube
Proposition $6$: Number Squared making Cube is itself Cube
Proposition $7$: Product of Composite Number with Number is Solid Number
Proposition $8$: Elements of Geometric Progression from One which are Powers of Number
Proposition $9$: Elements of Geometric Progression from One where First Element is Power of Number
Proposition $10$: Elements of Geometric Progression from One where First Element is not Power of Number
Proposition $11$: Elements of Geometric Progression from One which Divide Later Elements
Porism to Proposition $11$: Elements of Geometric Progression from One which Divide Later Elements
Proposition $12$: Elements of Geometric Progression from One Divisible by Prime
Proposition $13$: Divisibility of Elements of Geometric Progression from One where First Element is Prime
Proposition $14$: Expression for Integer as Product of Primes is Unique
Proposition $15$: Sum of Pair of Elements of Geometric Progression with Three Elements in Lowest Terms is Coprime to other Element
Proposition $16$: Two Coprime Integers have no Third Integer Proportional
Proposition $17$: Last Element of Geometric Progression with Coprime Extremes has no Integer Proportional as First to Second
Proposition $18$: Condition for Existence of Third Number Proportional to Two Numbers
Proposition $19$: Condition for Existence of Fourth Number Proportional to Three Numbers
Proposition $20$: For any finite set of prime numbers, there exists a prime number not in that set (Euclid's Theorem)
Proposition $21$: Sum of Even Integers is Even
Proposition $22$: Sum of Even Number of Odd Numbers is Even
Proposition $23$: Sum of Odd Number of Odd Numbers is Odd
Proposition $24$: Even Number minus Even Number is Even
Proposition $25$: Even Number minus Odd Number is Odd
Proposition $26$: Odd Number minus Odd Number is Even
Proposition $27$: Odd Number minus Even Number is Odd
Proposition $28$: Odd Number multiplied by Even Number is Even
Proposition $29$: Odd Number multiplied by Odd Number is Odd
Proposition $30$: Odd Divisor of Even Number also divides its Half
Proposition $31$: Odd Number Coprime to Number is also Coprime to its Double
Proposition $32$: Power of Two is Even-Times Even Only
Proposition $33$: Number whose Half is Odd is Even-Times Odd
Proposition $34$: Number neither whose Half is Odd nor Power of Two is both Even-Times Even and Even-Times Odd
Proposition $35$: Sum of Geometric Progression
Proposition $36$: Theorem of Even Perfect Numbers (first part)