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At Right Angles | Vol. 2, No. 3, November 2013 19 feature
Introduction
The French mathematician Pierre de Fermat (1601–1665) was a
veritable giant of number theory whose discoveries, and especially,
whose conjectures and unproven assertions, kept mathematicians
hard at work for several centuries that followed. Indeed, the �irst
two issues of this magazine both featured his work: the �irst issue
reviewed a book ([1]) on the history of what is known as “Fermat’s
Last Theorem,” while the second issue, in an article on the four
squares theorem ([2]), describes Fermat’s work on primes that are
representable as sums of two squares.
Besides these two well known contributions, Fermat is known for a
whole host of other theorems in mathematics. He was a lawyer by
training, but his passion was mathematics. He shone in arithmetic
(which in its more advanced form is what we call number
theory today), but made seminal contributions in other parts of
mathematics as well, and even in physics.
Fermat Numbers
A false conjecture leading to fun and fascination
Interspersed with historical and biographical details, this article has rich
nuggets of information. These don’t just exercise a student’s understanding of
exponents, they also provide solvable proofs for school students. Best of all, the
article weaves random results into a coherent whole, giving direction to ideas,
conjectures and proofs.
B. A. S���������
Learning from mistakes
Keywords: Fermat, Fermat number, Fermat prime, in�inity, regular polygons, constructibility
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20 At Right Angles | Vol. 2, No. 3, November 2013
Great mathematicians, and Fermat was squarely
in that league, are characterized by deep intuition
that enables them to see mathematical truths that
others are not yet able to see. But great
mathematicians are also human, and occasionally,
they are wrong. Fermat himself was wrong on at
least one mathematical matter: the issue of
whether numbers of the form 2��
+ 1 are prime.
These numbers are the subject of this article.
Recall �irst the convention when interpreting
numbers written with repeated exponents:
2��
+ 1 is to be interpreted as 2(��) + 1 (and not
(2�)� + 1). Let us write FF� for the number
2��
+ 1, so that FF� = 2��
+ 1 = 2� + 1 = 3, FF� = 5,
FF� = 17, etc. Fermat claimed that the numbers FF�
are prime for all integers nn nnn nn n . In fact, he
�irst claimed to have a proof, but later discovered
an error in it ([3, Foreward]). However, he
appeared to still believe in the truth of his claim.
It is thus fair to rename his claim as a conjecture.
Indeed, FF�, FF�, and FF� above are clearly prime. So
is FF� = 2� + 1 = 257 and FF� = 2�� + 1 = 65537.
But there the list is broken! Euler, who lived
approximately a century after Fermat
(1707–1783) showed that FF�, a ten-digit integer,
is not prime: it is divisible by 641. Thus, Fermat's
conjecture on the numbers 2��
+ 1 was false!
But there is another characterization of great
mathematicians that is relevant here—the very
objects they think about turn out to be fascinating
and deep, even if these mathematicians
occasionally make false assertions about them!
Such is indeed the case with numbers of the form
2��
+ 1, now appropriately called Fermat
Numbers. (Numbers of the form 2��
+ 1 that are
prime are now referred to as Fermat primes.)
Fermat numbers have many charming properties,
and have turned out to have intriguing
connections to other parts of mathematics, as well
as to computer science.
�dentities and t�e in�initude of primes
Let us start with some pretty identities that
Fermat numbers satisfy. Their proofs are fun
exercises for high school students, involving
nothing more than simple algebra and induction.
1. FF� = (FF��� − 1)� + 1, for nn n n.
2. FF� = FF� × FF� × FF� × ⋯ × FF��� + 2, for nn n n.
3. FF� = 2���� ⋅ FF� ⋅ FF� ⋅ FF� ⋅ ⋯ ⋅ FF��� + FF���, for
nn n n.
4. FF� = FF�
��� − 2(FF��� − 1)�, for nn n n.
We will prove the �irst one here: Note that
2��
= 2������ = 2����⋅� = (2���� )� = (FF��� − 1)�.
Adding one everywhere, we �ind
FF� = 2��
+ 1 = (FF��� − 1)� + 1, as desired.
There is an immediate consequence of the second
identity above: the last digit of every Fermat
number (for nn n n) must be 7. This is because for
nn n n, we have
FF� = 3⋅5⋅FF� ⋅ ⋯ ⋅FF���+2 = 5(3⋅FF� ⋅ ⋯ ⋅FF���)+2.
So FF� is of the form 2 plus an odd multiple of 5
and hence has last digit 7. Pretty!
The second consequence is that the Fermat
numbers are pairwise relatively prime; that is, for
distinct non-negative integers ii and jj,
gcd(FF�, FF�) = 1. This is attributed to Christian
Goldbach (who is well known for a conjecture that
is as yet unproven: Every even integer greater
than 2 is expressible as a sum of two primes). As
noted in a companion article in this issue, There
are �n�initel� man� �rimes, this property leads to
another proof of the in�initude of primes.
Fermat numbers and constructibility of
polygons
Recall the problems of constructibility handed to
us by the Greeks: using only straight-edge and
compass, construct line segments of speci�ied
lengths, and angles of speci�ied measures. It was
an open problem for a very long time, for
instance, (i) whether one could trisect an
arbitrary angle using straight-edge or compass,
(ii) whether one could “square the circle,” that is,
construct the side of a square whose area is that
of a given circle, and (iii) whether one could
“double the cube,” that is, construct the side of a
cube whose volume is twice that of a given cube.
These problems are easy to solve once one has at
one's command techniques from Field Theory
(known earlier as the ‘Theory of Equations’); but
this theory was not known to the Greeks. We now
know that the answer all three questions is: No!
A speci�ic problem in this context was the
constructibility of regular nn-gons for various
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At Right Angles | Vol. 2, No. 3, November 2013 21
values of nn. Thus, a regular 3-gon is an equilateral
triangle, a regular 4-gon is a square, a regular
5-gon is a regular pentagon, and so on. Whether a
regular nn-gon can be constructed using a
straight-edge and compass quickly reduces to the
question of whether the angle 360∘
/nn can be
constructed using straight-edge and compass.
This problem was investigated by the Great
Master, Carl Friedrich Gauss. (Gauss ranks among
the greatest mathematicians ever, when
measured not just by his own productivity but by
the new areas of mathematics he initiated; his
results to this date are a source of joy and wonder.
His in�luence on mathematics and indeed all
sciences ranks with that of Newton.) Gauss
showed that the regular 17-gon is constructible
(note that 17 is FF�), and went on to show that a
regular nn-gon (nn n n) can be constructed if the
prime factorization of nn is of the form
2�pp�pp� ⋯ pp�, where the pp� are distinct primes of
the form 2��
+ 1; Fermat numbers again! This is
an instance of how questions about objects
considered by great mathematicians (in this case
Fermat) can turn out to have deep signi�icance in
mathematics, far from apparent at �irst. Thus, the
question of whether for a given kk the kkth Fermat
number 2��
+ 1 is prime turns out to be more
than just a curiosity: it is vitally connected to
whether an nn-gon can be constructed.
The reason why a regular nn-gon with the stated
prime factorization of nn is constructible, lies in
Field Theory. For the case where nn is a
prime—call it pp instead—the theory tells us that
the regular pp-gon is constructible if and only if
pp p p is a power of 2. Thus, a regular pp-gon is
constructible if and only if pp p p� + 1 for some
integer kk.
Now one can see quite easily that 2� + 1 cannot be
prime unless kk is itself a power of 2. For, suppose
kk kk�
bb for some odd integer bb b b. Then
2� + 1 = 2��
� + 1 = (2��
)� + 1.
Now it is a fact (it can be proven as a high school
exercise) that xx� + 1 is divisible by xx xx if bb is
odd. Hence if bb b b, pp p pp��
)� + 1 would have
the strictly smaller divisor 2��
+ 1, contradicting
the fact that pp is prime. Hence, the condition from
Field Theory becomes: for prime pp pp, a regular
pp-gon is constructible if and only if pp is a Fermat
prime!
The condition for the constructibility of a regular
nn-gon for a general nn follows from the condition
just described for the case where nn is prime, using
standard reductions also furnished by Field
Theory. Indeed, the condition for a general nn-gon
is also an if and only if statement: a regular nn-gon
is constructible if and only if nn is of the form
described by Gauss. Gauss proved the ‘if’ part of
the condition, but his proof of the ‘only if’ part had
a gap that was �illed only later ([�, Chap. ��]).
Primality of the Fermat numbers
Let us turn to the original conjecture of Fermat,
that the numbers 2��
+ 1 are prime for all
nn nnn nn n . We know, thanks to Euler, that while
FF� through FF� are prime, FF� is not. For what other
values of nn is FF� known to be prime? The answer,
more than three hundred and �ifty years after
Fermat made his �irst conjecture, is: None!
That does not mean that no FF� is prime for nn n n.
All it means is that no one has as yet found a
prime FF� for nn n n. What has been established
are many results in the opposite direction (similar
to the case of FF�): the numbers FF� through FF��
have all been shown to be composite ([4]).
Besides these, FF� is known to be composite for
other sporadic values of nn, such as
nn nnnn nnn nnn nnnn nnnnn nnnnn nnnnn, to select
just a sample ([4]).
What makes determination of the primality of FF�
so dif�icult is that, thanks to the presence of the
double exponent, the number of digits in FF�
grows very rapidly as nn becomes large. In fact,
Exercise (2) shows that the growth in the number
of digits is exponential.
On the other hand, there is a very pretty result on
the possible prime factors of FF�: Euler showed
that any prime that divides FF� must be of the
form kk k k��� + 1, for some positive integer kk.
(The proof of this itself involves another famous
theorem of Fermat known as Fermat's Little
Theorem: for any prime pp and any integer aa, the
number aa� − aa is divisible by pp.) Euler's result
was further sharpened by Lucas, who showed