Euclid's proof of the infinitude of primes uses mathematical induction. It considers any finite list of prime numbers p1, p2, ..., pn and constructs the number q = P + 1, where P is the product of the primes in the list. Euclid shows that either q is prime, proving there are more primes than in the list, or some prime factor of q cannot be in the list, again showing there are more primes. This proves there is no largest prime number and primes are infinite. Later mathematicians like Euler, Erdos, and Furstenberg provided alternative proofs using unique prime factorizations and divergent series.

1. Name :- karan balchandani
Std :- 10th
Subject :- maths
Topic :- euclid’s theorem

2. 1 Euclid's proof
2 Euler's proof
3 Erdős' proof
4 Furstenberg's proof
5 Some recent proofs
5.1 Pinasco
5.2 Whang
6 Proof using the irrationality of π

3. Euclid's theorem is a fundamental statement in
number theory that asserts that there are infinitely
many prime numbers. There are several well-
known proofs of the theorem.

4. Euclid offered the following proof published in his work Elements (Book IX, Proposition 20),[1] which is paraphrased
here.
Consider any finite list of prime numbers p1, p2, ..., pn. It will be shown that at least one additional prime number
not in this list exists. Let P be the product of all the prime numbers in the list: P = p1p2...pn. Let q = P + 1. Then q is
either prime or not:
If q is prime, then there is at least one more prime than is in the list.
If q is not prime, then some prime factor p divides q. If this factor p were on our list, then it would divide P (since
P is the product of every number on the list); but p divides P + 1 = q. If p divides P and q, then p would have to divide
the difference[2] of the two numbers, which is (P + 1) − P or just 1. Since no prime number divides 1, this would be a
contradiction and so p cannot be on the list. This means that at least one more prime number exists beyond those in
the list.

5. This proves that for every finite list of prime numbers
there is a prime number not on the list, and therefore
there must be infinitely many prime numbers.
Euclid is often erroneously reported to have proved
this result by contradiction, beginning with the
assumption that the set initially considered contains
all prime numbers, or that it contains precisely the n
smallest primes, rather than any arbitrary finite set of
primes.[3]
Although the proof as a whole is not by
contradiction (it does not assume that only finitely
many primes exist), a proof by contradiction is within
it, which is that none of the initially considered primes
can divide the number q above.

6. Another proof, by the Swiss mathematician Leonhard Euler, relies on the
fundamental theorem of arithmetic: that every integer has a unique prime
factorization. If P is the set of all prime numbers, Euler wrote that:
The first equality is given by the formula for a geometric series in each term of
the product. To show the second equality, distribute the product over the sum:
in the result, every product of primes appears exactly once and so by the fundamental
theorem of arithmetic the sum is equal to the sum over all integers.
The sum on the right is the harmonic series, which diverges. Thus the product on the left
must also diverge. Since each term of the product is finite, the number of terms must be
infinite; therefore, there is an infinite number of primes.

7. Paul Erdős gave a third proof that relies on the fundamental theorem
of arithmetic. First note that every integer n can be uniquely written as
where r is square-free, or not divisible by any square numbers (let s2 be the largest square
number that divides n and then let r=n/s2). Now suppose that there are only finitely many
prime numbers and call the number of prime numbers k.
Fix a positive integer N and try to count the number of integers between 1 and N. Each of
these numbers can be written as rs2 where r is square-free and r and s2 are both less than
N. By the fundamental theorem of arithmetic, there are only 2k square-free numbers r (see
Combination#Number of k-combinations for all k) as each of the prime numbers factorizes r
at most once, and we must have s<√N. So the total number of integers less than N is at
most 2k√N; i.e.:

8. In the 1950s, Hillel Furstenberg introduced a proof
using point-set topology. See Furstenberg's proof
of the infinitude of primes.

9. Juan Pablo Pinasco has written the following proof.[4]
Let p1, ..., pN be the smallest N primes. Then by the inclusion–exclusion principle, the
number of positive integers less than or equal to x that are divisible by one of those
primes is
Pinasco

11. Euclid
Greek mathematician – “Father
of Geometry”
Developed mathematical proof
techniques that we know today
Influenced by Plato’s
enthusiasm for mathematics
On Plato’s Academy entryway:
“Let no man ignorant of
geometry enter here.”
Almost all Greek
mathematicians following
Euclid had some connection
with his school in Alexandria

12. Euclid’s Elements
Written in Alexandria around 300 BCE
13 books on mathematics and geometry
Axiomatic: began with 23 definitions, 5 postulates,
and 5 common notions
Built these into 465 propositions
Only the Bible has been more scrutinized over time
Nearly all propositions have stood the test of time

13. Preliminaries: Definitions
Basic foundations of Euclidean geometry
Euclid defines points, lines, straight lines, circles,
perpendicularity, and parallelism
Language is often not acceptable for modern
definitions
Avoided using algebra; used only geometry
Euclid never uses degree measure for angles

14. Preliminaries: Postulates
Self-evident truths of
Euclid’s system
Euclid only needed five
Things that can be done
with a straightedge and
compass
Postulate 5 caused some
controversy

15. Preliminaries: Common Notions
Not specific to geometry
Self-evident truths
Common Notion 4: “Things which coincide with one
another are equal to one another”
To accept Euclid’s Propositions, you must be satisfied
with the preliminaries

16. Early Propositions
Angles produced by
triangles
Proposition I.20: any two
sides of a triangle are
together greater than the
remaining one
This shows there were
some omissions in his
work
However, none of his
propositions are false
Construction of triangles
(e.g. I.1)

17. Early Propositions: Congruence
SAS
ASA
AAS
SSS
These hold without reference to the angles of a
triangle summing to two right angles (180˚)
Do not use the parallel postulate

18. Parallelism and related topics
Parallel lines produce
equal alternate angles
(I.29)
Angles of a triangle sum to
two right angles (I.32)
Area of a triangle is half
the area of a parallelogram
with same base and height
(I.41)
How to construct a square
on a line segment (I.46)

19. Pythagorean Theorem: Euclid’s
proof
Consider a right triangle
Want to show a2
+ b2
= c2

20. Pythagorean Theorem: Euclid’s
proof
Euclid’s idea was to use areas of squares in the proof.
First he constructed squares with the sides of the
triangle as bases.

21. Pythagorean Theorem: Euclid’s
proof
Euclid wanted to show that the areas of the smaller
squares equaled the area of the larger square.

22. Pythagorean Theorem: Euclid’s
proof
By I.41, a triangle with the same base and height as
one of the smaller squares will have half the area of
the square. We want to show that the two triangles
together are half the area of the large square.

23. Pythagorean Theorem: Euclid’s
proof
When we shear the triangle like this, the area does
not change because it has the same base and height.
Euclid also made certain to prove that the line along
which the triangle is sheared was straight; this was
the only time Euclid actually made use of the fact that
the triangle is right.

24. Pythagorean Theorem: Euclid’s
proof
Now we can rotate the triangle without changing it.
These two triangles are congruent by I.4 (SAS).

25. Pythagorean Theorem: Euclid’s
proof
We can draw a perpendicular (from A to L on
handout) by I.31
Now the side of the large square is the base of the
triangle, and the distance between the base and the
red line is the height (because the two are parallel).

26. Pythagorean Theorem: Euclid’s
proof
Just like before, we can do another shear without
changing the area of the triangle.
This area is half the area of the rectangle formed by
the side of the square and the red line (AL on
handout)

27. Pythagorean Theorem: Euclid’s
proof
Repeat these steps for the triangle that is half the area
of the other small square.
Then the areas of the two triangles together are half
the area of the large square, so the areas of the two
smaller squares add up to the area of the large square.
Therefore a2
+ b2
= c2
!!!!

28. Pythagorean Theorem: Euclid’s
proof
Euclid also proved the converse of the Pythagorean
Theorem; that is if two of the sides squared equaled
the remaining side squared, the triangle was right.
Interestingly, he used the theorem itself to prove its
converse!

29. Other proofs of the Theorem
Mathematician Proof
Chou-pei Suan-ching
(China), 3rd
c. BCE
Bhaskara (India),
12th
c. BCE
James Garfield (U.S.
president), 1881

30. Further issuesControversy over parallel postulate
Nobody could successfully prove it
Non-Euclidean geometry: Bolyai, Gauss, and
Lobachevski
Geometry where the sum of angles of a triangle is less
than 180 degrees
Gives you the AAA congruence