Euclid's theorem on the infinite nature of primes has captivated ma...
Assuming there are only finitely many primes, the product of all th...
The author of the paper is Romeo Meštrović. Romeo is a Full Profess...
AVeryShortProofoftheInfinitudeofPrimes
If we suppose that the number of primes is not infinite, then by using an “even-odd”
argument, we prove that 3 is a greatest prime, a contradiction.
Theorem. (Euclid [1], [3,p.3])Thereareinfinitelymanyprimes.
Euclid’s famous theorem on the infinitude of primes has fascinated generations
of mathematicians since its first and famous demonstration given by Euclid (300
B.C.) in his celebrated Elements ([1]BookIX,Proposition20).Evenafteralmost
two and a half millennia, Euclid’s theorem on the infinitude of primes stands as an
excellent model of reasoning. In [2]theauthorofthisnoteprovidedacomprehensive
historical survey of different proofs of Euclid’s theorem on the infinitude of primes.
In this note, we give a short elementary proof of Euclid’s theorem as follows.
Proof. Suppose that p
1
= 2 < p
2
= 3 < p
3
< ··· < p
k
are all the primes. Consider
the product P = 3 p
3
p
4
··· p
k
.Thenbytheassumptionitfollowsthat{1, 2, 2
2
,...,
2
n
,...} is a set of all positive integers that are relatively prime to P.Inparticular,
since 2 is relatively prime to P,itfollowsthatP − 2 is also relatively prime to P.
Therefore, P − 2isanoddintegerbelongingtotheset{1, 2, 2
2
,...,2
n
,...}.This
shows that P − 2 = 1, i.e., P = 3. This means that 3 is a greatest prime number. A
contradiction, and the proof is completed.
Remark. Notice that our proof works if we replace the prime p
1
= 2byanyprime
p
j
with j ∈{2, 3,...,k − 1}.Thenassumingtheanalogousargumentasinthe
proof above, we arrive at the equation p
1
p
2
··· p
j−1
p
j+1
··· p
k
− p
j
= 1, which
is impossible because the left-hand side of the previous equation is greater than
2 p
k
− p
j
> p
j
> 1. Observe also that this argument cannot be applied for the
“greatest prime” p
k
instead of p
j
;namely,for j = k we find that p
1
p
2
··· p
k−1
=
p
k
+ 1. However, we can see from several number theory arguments that the previous
equation is satisfied only for k = 3(thatis,2· 3 = 5 + 1).
REFERENCES
1. T. L. Heath, The Thirteen Books of Euclid’s Elements. Vol. 2. Second ed. Cambridge Univ. Press,
Cambridge, 1908, reprinted by Dover, New York, 1956.
2. R. Me
ˇ
strovi
´
c, Euclid’s theorem on the infinitude of primes: A historical survey of its proofs (300
B.C.–2012), 66 pages, preprint, http://www.arXiv:1202.3670v2[math.HO],2012.
3. P. Ribenboim, The Little Book of Bigger Primes. Second ed. Springer-Verlag, New York, 2004.
—Submitted by Romeo Me
ˇ
strovi
´
c, University of Montenegro
http://dx.doi.org/10.4169/amer.math.monthly.124.6.562
MSC: Primary 11A41
562
c
⃝ THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 124
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Assuming there are only finitely many primes, the product of all the odd primes (P) is an odd number and greater than 3.
The positive integers that are co-prime to P are precisely the powers of 2. Now we note that P-2 is co-prime to P (any common divisor would divide 2, and 2 does not divide P) so P-2 must be a power of 2. Since P is odd, that implies P-2=1 and P=3, which is a contradiction - P is not the greatest prime.
Euclid's theorem on the infinite nature of primes has captivated mathematicians across generations ever since its famous proof by Euclid in 300 B.C.
Paul Erdös famously said “It will be another million
years, at least, before we understand the primes” and yet this has not deterred mathematicians throughout the centuries to explore the mysteries of primes and their unique properties.
Many renowned mathematicians from the 18th and 19th centuries, such as Goldbach, Euler, Lebesgue, Kronecker, Hensel, Kummer, Stieltjes, and Hermite, contributed various proofs of the infinitude of primes. Additionally, in the past century, notable mathematicians including I. Schur, K. Hensel, G. Pólya, Erdö and G. H. Hardy, among others, have offered further compelling proofs, including demonstrations of the infinitude of primes in different arithmetic progressions.
Some of these proofs have already been annotated on Fermat's Library including [Harry Furstenberg's](https://fermatslibrary.com/s/on-the-infinitude-of-primes) and [Sam Northshield's one-line proof](https://fermatslibrary.com/s/a-one-line-proof-of-the-infinitude-of-primes).
There is no way to determine how many different proofs of the infinitude of primes exist and how many will emerge in coming years. In this one, published in the July 2017 issue of the American Mathematical Monthly, the author uses an "even-odd" argument to create a very short proof of the infinitude of primes.
The author of the paper is Romeo Meštrović. Romeo is a Full Professor of Mathematics at the Faculty of Maritime Studies in Kotor, University of Montenegro. He was awarded the Bronze Medal in the XXI International Mathematical Olympiad in London in 1979. His research areas include Complex and Functional Analysis, Combinatorial and Elementary Number Theory, and applications of Queuing Theory in modeling port and terminal operations.
P is the product of all odd primes, rather than the greatest prime. Since 5 is a prime, we know P ≥ 15, which contradicts P=3.