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Sarah Yeust
Honors Number Theory
November 2011
The Never-Ending Questions about Prime Numbers
The Puzzles of Prime Numbers
It seems that as soon as students learn the definition of a prime number, their next
question is, “How many prime numbers exist?” No sooner have they identified 3, 5, and 7 as
prime numbers than their minds are shuffling to calculate the largest prime number they know.
Questions concerning prime numbers are not new to the study of Mathematics and Number
Theory. Although modern technology allows computation and numeric calculations to take place
at an extraordinarily fast pace, the study of prime numbers extends back in history, even reaching
to the times of the Greek civilizations. For centuries, mathematicians have grappled with issues
concerning prime numbers, and the progression made is both remarkable but also indicative of
the discoveries still to come.
Several major questions have propelled the study of prime numbers. First,
mathematicians and researchers from other branches of science have asked how many prime
numbers exist. If numbers are considered to continue to infinity, does this mean that there is an
infinite amount of prime numbers as well? Secondly, another major issue has been how to
recognize whether a natural number is a prime number. Are there tests or models that can be
followed to determine whether any number, selected at random, is a prime number? Another
common question surrounding prime numbers concerns their distribution. Mathematicians
contemplate the gaps between successive prime numbers and whether these gaps are coincidental
or significant (Ribenboim). All of these inquiries are only the beginning of the work that has
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been dedicated to prime numbers, but the discoveries made so far provide the foundation for the
work that continues to be carried out in labs and universities across the globe today.
How Many Prime Numbers Exist?
The concept of a prime number is one of the most basic topics in the study of Number
Theory. A given integer, p, is prime if p ≠ ±1 and the equation p = ab has no solution in integers
a and b except for those for which a = ±1 or a = ±p (LeVeque Fundamentals 2). Stated another
way, a prime number is any integer greater than 1 that has only 1 and itself as exact divisors. It is
important to note that by conventional agreement, 1 is not a prime number (Ogilvy 146).
Furthermore, a number greater than 1 that is not prime is called composite (Hardy 2).
Euclid’s reasoning has been a model for many topics of study throughout the history of
Mathematics. Largely regarded as Euclid’s First Theorem, Euclid proposed that if p is a prime
and p | ab , then p | a or p | b where | means divide. Euclid’s second theorem states that the
number of primes is infinite. This theorem was proved by Euclid in his book, Elements
(Proposition IX.20). His general argument was as follows: Suppose p is the largest prime. Then
if p and all the smaller primes are multiplied together and 1 is added to the product, the new
number cannot be divisible by p or any lesser prime. It must therefore be itself a prime or
divisible by some prime greater than p. In either case, there is a prime greater than p. More
formally:
Proof 1: Euclid’s Proof of the Infinitude of Primes
Suppose that p1 = 2 < p2 = 3 < … < pr are all the primes. Let P = p1p2…pr + 1 and let p be
a prime dividing P; then p cannot be any of p1, p2, …pr, otherwise p would divide the
difference p – p1p2…pr =1, which is impossible. So this prime p is still another prime, and
p1, p2, … pr would not be all the primes.
▄
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Euclid’s proof is simple, but it does not give any information about the new prime. The
only fact shown is that the new prime is at most equal to the number P, but it may also be smaller
(Ribenboim 4). Also, it does not investigate the occurrence of primes or how regularly (or
irregularly) they appear. Other mathematicians followed Euclid, providing their own variations
of proofs for the infinitude of prime numbers.
In 1878, German mathematician Ernst Eduard Kummer gave the following proof for the
infinitude of prime numbers:
Proof 2: Kummer’s Proof of the Infinitude of Primes
Suppose that there exist only finitely many primes p1 < p2 < …< pr. Let N = p1p2…pr > 2.
The integer N-1, being a product of primes, has a prime divisor pi in common with N; so,
pi divides N-(N-1) =1, which is absurd!
▄
Kummer’s proof was very succinct and simple. It was also similar to Euclid’s work since it was a
proof by contradiction which failed due to the conclusion that some prime must divide one
(Ribenboim 4-5).
Euler (1707-1783), largely credited for developing the foundation of mathematical notation
in use today (Boyer 441), also provided a proof of the infinitude of primes, though it was rather
indirect and in some sense, unnatural. Euler “showed that there must exist infinitely many primes
because a certain expression formed with all the primes is infinite” (Ribenboim 7). For example,
if p is any prime, then 1/p < 1; the sum of the geometric series is
∞
1
∑p
k =0
k
=
1
.
1 − (1 / p )
Similarly, if q is another prime, then
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∞
1
∑q
k =0
k
=
1
.
1 − (1 / q )
When these equalities are multiplied:
1+
1 1 1
1
1
1
1
.
+ + 2 +
+ 2 + ... =
×
p q p
pq q
1 − (1 / p ) 1 − (1 / q)
The left side, which is the sum of the inverses of all the natural numbers of the form phqk
( h ≥ 0, k ≥ 0) , each counted only once because every natural number has a unique factorization
that is the product of primes. This idea is the basis of the proof (Ribenboim 7-8).
Proof 3: Euler’s Proof of the Infinitude of Primes
Suppose that p1, p2, … , pn are all the primes. For each i = 1, … n
∞
1
∑p
k =0
k
i
=
1
.
1 − (1 / pi )
Multiplying these n equalities, one obtains
∞ 1 
∑
 =
∏
k
i =1  k = 0 p i


m
m
1
∏ 1 − (1 / p )
i =1
,
i
and the left-hand side is the sum of the inverses of all natural numbers, each counted
once – this follows from the fundamental theorem that every natural number is equal, in a
unique way, to the product of primes.
∞
But the series
1
∑  n  is divergent; being a series of positive terms, the order of
n =1
summation is irrelevant, so the left- hand side is infinite, while the right-hand side is
clearly finite. This is absurd.
▄ (Ribenboim 7-8).
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In 1980, Washington provided a proof that is largely based on commutative algebra and
ideas of principal ideal domains, unique factorization domains, Dedekind domains, and algebraic
numbers (Ribenboim 10-11). In this context, “domain” refers to an integral domain which is a
ring that is commutative under multiplication, has an identity element, and no divisors of 0
(Dummit and Foote 228). In every finite number field, the Dedekind domain is the ring of
algebraic integers. Every ideal is the product of prime ideals. Also, in every finite number field,
there are finitely many prime ideals that divide any given number. A principal ideal domain is a
Dedekind domain with only finitely many primes ideals, and every element is the product of
prime elements in a unique way (Ribenboim 11).
Proof 4: Washington’s Proof of the Infinitude of Primes
Consider the field of all numbers of the form a + b − 5 where a, b are rational numbers.
The ring of algebraic integers in this field consists of the numbers of the above form, with
a, b ordinary integers. It is easy to see that 2, 3, 1 + − 5 , 1 − − 5 are prime elements of
this ring, since they cannot be decomposed into factors that are algebraic integers, unless
(
)(
)
one of the factors is a “unit” 1 or -1. Note also that since 1 + − 5 1 − − 5 = 2 x 3, the
decomposition of 6 into a product of primes is not unique up to units, so this ring is not a
unique factorization domain; hence, it is not a principal ideal domain. So it must have
infinitely many prime ideals, and so there exist infinitely many prime numbers
(Ribenboim 11).
▄
Another remarkable theorem connected with the infinitude of primes was proven by Peter
Gustav Lejeune Dirichlet in 1837. Dirichlet was a successor of Gauss at Göttingen, and the
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theorem he proved can be considered a deep and difficult corollary to Euclid’s theorem on the
infinity of primes (Boyer 502). This theorem is known as Dirichlet’s Theorem and states that if
(a, d) = 1 with a > 0 and d > 0, then there are infinitely many primes of the form a + kd. The
conditions of Dirichlet’s theorem are necessarily true since if (a, d) = r >1, then r|(a + kd) for
every k and a + kd is never a prime for k ≥ 1 . The proof of Dirichlet’s Theorem is too extensive
for this paper, but the theory all rests in the infinitude of primes (Long 72-74).
The proofs above are just a small selection of the countless that have been provided by
mathematicians throughout the decades. Interestingly, most approach the problem in slightly
different ways. However, relatively little information can be gained from these proofs concerning
the significance of any arbitrary prime number or even how to determine whether a number is
prime. For these reasons, the proofs concerning the infinitude of prime numbers are only the
starting point for prime number studies.
How does one recognize whether a natural number is prime or composite?
Stemming from the proofs of the infinitude of prime numbers, the next issue to be
explored is how to determine whether or not a natural number is a prime number. Factorization
and primality have become important in public key cryptography, and this is one of the most
striking applications of Number Theory (Ribenboim 13-14). No feasible answer has ever been
given to the question of how to determine whether a number is prime or composite. However, the
work of mathematicians has yielded two types of solutions: impractical and complete or practical
but incomplete (Stewart 56). An example of a solution that is impractical but complete is the
“sieve of Eratosthenes.” As a contributor to many fields of learning, it is no surprise that
Eratosthenes developed a procedure for isolating the prime numbers, termed the “sieve of
Eratosthenes.” Beginning with all the natural numbers (see Table 1) arranged in order, one
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strikes out every second number after the number two. Next, following the original sequence,
one strikes out every third number following the number three, every fifth number following the
number five, and continues to strike out every nth number following the number n. The
remaining numbers, from 2 on, will be prime numbers (Boyer 161). This procedure is done in
the table below for the numbers one through fifty. The method became the way to provide
estimates for the number of primes satisfying given conditions (Ribenboim 15).
Table 1
(Sieve of
Eratosthenes)
A variant on the sieve provides immediate answers of primality (see Table 2). The first
row and column consist of the arithmetic progression where a = 4 and d = 3, where a is the first
term and d is the common difference. The rest of the rows and columns have their a’s fixed, and
for d, use 5 for the second row and column, 7 for the third row and column, and so on. The
following criteria now follow: If x is any integer greater than 2, x is prime if and only if
does not appear in the table (Ogilvy 98).
x −1
2
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Table 2
4
7
10
13
16
.
7
12
17
22
27
.
10
17
24
31
38
.
13
22
31
40
49
.
16
27
38
49
60
.
.
.
.
.
.
.
(Ogilvy 98)
In one method to prove this claim, let n =
x −1
, and then x = 2n+1. Even though the
2
theorem states that x is composite if and only if n appears in the table, the focus of the problem is
not the number but twice its value plus one. A comparison table would be as follows, where each
number is replaced by twice its value plus one:
Table 3
(Ogilvy 99)
9
15
21
27
33
.
15
25
35
45
55
.
21
35
49
63
77
.
27
45
63
81
99
.
33
55
77
99
121
.
.
.
.
.
.
.
Observe that the even numbers are all omitted. The first row and column contain all the
odd multiples of 3 except 3 itself, and the second row and column list all the odd multiples of 5
except 5 itself, and so on. The table lists all and only the odd composite numbers, even including
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some of them more than once (for example: 105 = 3 x 5 x 7). All even primes will be excluded,
since the table omits all even composites. However, since 2 is the only even prime, there must be
a stipulation added to the theorem that x must be greater than 2 itself, since 2 is not included on
the table (Ogilvy 99-100).
Another major contributor to work on prime numbers was Fermat. By 1629, Fermat was
making important discoveries in Mathematics. His background in Mathematics, the sciences, and
literature provided him with a good foundation for his mathematical work; he is often considered
the only rival of Descartes (Boyer 346). The fundamental property of prime numbers was
expressed as a special case by the Chinese over 2,000 years ago but is now known as Fermat’s
(lesser) Theorem. It states that “If n is any integer and p is a prime, then np- n is exactly divisible
by p (in other words, n p − n ≡ 0(mod p) ). Fermat first declared the general form in a letter to
another mathematician in 1640. The more common form of this theorem reads “If p is a prime
number, and n an integer not a multiple of p, then np-1 -1 is exactly divisible by p.” Today, the
proof of this takes only a few lines, but Leibniz (1646-1716) proposed the first proof of this
during a time when algebraic symbolism was in a primitive state (Taylor 70).
In 1950, G. Giuga considered the converse of Fermat’s (lesser) Theorem, questioning
whether it was true. By Fermat’s little theorem, then if p is prime,
1 p −1 + 2 p −1 + ... + ( p − 1) p −1 ≡ −1(mod p ) . Thus:
1 + 1 + ... + 1 ≡ ( p − 1) ≡ −1 .
Thus, is the converse true? If n > 1 and n divides 1n-1 + 2n-1 + …+ (n-1)n-1 +1, then does
this mean that n is a prime number? No. It follows that n satisfies Giuga’s condition if and only
if, for every prime p dividing n, p2(p-1) divides n-p (Ribenboim 20-21). Informally, Giuga even
had a number named after him. “Any composite number n with p|(n/p-1) for all prime divisors p
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of n” is called a Giuga Number (“Giuga’s” 2). For example, 30 is a Giuga number. The prime
factors of 30 are 2, 3, and 5. We can verify that 30/2 -1 =14, which is divisible by 2. Also,
30/3 -1 = 0, which is divisible by 3, and finally, 30/5 -1 =5, which is the third prime factor.
Therefore, 30 meets the conditions and is a Giuga number.
Previously, in 1862, English mathematician Joseph Wolstenholme proved that if p is a
prime, p ≥ 5, then the numerator of
1+
1 1
1
+ + ... +
is divisible by p2, and the numerator of
2 3
p −1
1+
1
1
1
+ 2 + ... +
is divisible by p.
2
2
3
(1 − p )2
Along the same lines of logic, Henry Mann and Daniel Shanks (1972) characterized
prime numbers by divisibility of binomial coefficients: p is a prime number if and only if, for
every
 p + 2  p + 2
 p
, 
+1, …,   , k divides
k= 


 3   3 
2
k
.
p-2k
This notation means that for any real number x, [x ] indicates the only integer such that
[x] ≤ x < [x] + 1. Therefore, [x] is called the largest integer x (Ribenboim 21).
The works of Giuga, Wolstenholme, Mann and Shanks all illustrate properties of primes
that were formalized by theorems and proofs (Ribenboim 20-21).
Another important theorem about prime numbers was published by English
mathematician Edward Waring (1736-1798) and named after his friend, Josh Wilson (17411793). This theorem states that “If p is a prime, then (p - 1)! +1 is a multiple of p” (Boyer 458).
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This theorem is reminiscent of Euclid’s proof since it concerns the possibility of numbers which
divide one, but it is contrary to Fermat’s theorem since it is true only when p is prime (Taylor
70).
French monk Father Marin Mersenne is largely credited with starting the chain of events
that led to further studies on the identification of prime numbers. In 1644, he found that the first
few primes p for which Mp = 2p-1 is prime are 2, 3, 5, 7, 13, and 19. He conjectured that Mp
would be prime for p = 31, 67, 127, and 257 and that no other primes for p would occur in this
range. Later, Mersenne was found to be mistaken; M61 , M89 , and M107 are prime, and M67 and
M257 are not. However, in his honor, numbers in the form of 2n-1 are called Mersenne primes. By
1963, the list of Mersenne primes had grown to 23, and additions included p = 521, 607, 1279,
2203, 2281, 3217, 4253, 4423, 9689, 9941, and 11,213. Between 1974 and 1994, several other
additions were added, and in 1994, M859,433 was added after several hours of work on a
supercomputer. Eventually, a primality test was developed for determining whether a Mersenne
number was prime, since there was a vast amount of computer memory needed (Long 81-82).
Fermat continued these concepts and conjectured that 2 2 + 1
n
is a prime for every
nonnegative integer n. While this is true for n = 0, 1, 2, 3, and 4, in 1732, Euler showed that
2 2 + 1 is divisible by 641 and thus, is not a prime. No more Fermat primes have been
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discovered, and a fraction of mathematicians believe that no others exist (Long 82).
Euler introduced a generalization of Fermat’s little theorem in 1760. He began by
examining those integers which are relatively prime to one another (have no common factors).
For example, 7, 8, 9, and 25 are all relatively prime even though three of them are composite.
Any number m has so many integers less than m which are relatively prime to it. The number of
such integers is denoted by φ (m ) , known as Euler’s function. When m is composite there is
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counting that needs to be completed. For example, when m = 14, three are six integers prime to
m – 1, 3, 5, 9, 11, and 13, and therefore, φ (14) = 6. However, when m is a prime, then clearly
φ ( m ) = m − 1 . Euler substitutes this for p-1 in Fermat’s statement that np-1-1 is divisible by p, and
this theorem includes all values of m. Thus, it now becomes
n
n φ ( m ) −1 is exactly divisible by m,
provided only that n is relatively prime to mn (Taylor 72).
Work with prime numbers also branches into modulo arithmetic. By definition, if m is
positive and m | (a-b), we say that a is congruent to b modulo m, which we write as
a ≡ b(mod m ) . If a is not congruent to b modulo m, we write a ≡ b(mod m ) (Long 88). The
converse of Fermat’s little theorem was found to be very useful and true. It was discovered by
Lucas in 1876 and states: Let N>1. Assume that there exists an integer a>1 such that
i) aN-1 ≡ 1 (mod N)
ii) am ≡ 1 (mod N), for m = 1, 2, …, N-2.
Then N is prime.
Unfortunately, this test requires N-2 successive multiplications by a, and finding residues
modulo N. Therefore, this requires too many calculations.
Lucas gave the following test in 1891:
Let N >1. Assume that there exists an integer a>1 such that:
i. aN-1 ≡ 1 (mod N)
ii. am ≡ 1 (mod N), for every m < N, such that m divides N -1.
Then N is prime.
Unfortunately, this test requires the knowledge of all factors N-1; thus, it is only easily
applicable when N has a special form, such as N = 2n+1 or N = 3 x 2n+1.
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Finally, Derrick Henry Lehmer made Lucas’s tests more practical in 1927, which was
then made even more flexible in 1975 by John Brillhart and J. L. Selfridge. The test now read:
Let N > 1. Assume that for every prime factor q of N-1 there exists an integer a = a (q) >1
such that
i. aN-1 ≡ 1(mod N )
ii. a
( N −1)
q
≡ 1(mod N )
Then N is a prime.
While it is still necessary to know the prime factors of N-1, fewer congruences have to be
satisfied (Ribenboim 37-38).
Finally, there are several other miscellaneous criteria that have been suggested for prime
numbers . While they have not all been proved, they give an idea of concentrated effort and the
diversity of the approach that has been directed toward the elusive prime numbers.
a)A number is prime if it is not expressible in the form ab+xy when a, b, x, y are positive
integers such that a+b = x-y.
b)If p is a prime of the form 4k+1, then 2p+1 is also a prime if it exactly divides (2p+1)/3.
c)A number of the form 6n-1 is prime if n is not of the form 6xy+x-y.
d) At least four primes lie between the squares of two consecutive primes (>3).
e)If n is an odd number, then 4n+1 is prime if it is a factor of (22n+1)/5.
Nowadays, mathematicians do not search for a formula that exclusively generates prime
numbers. A more reasonable question is, “What is the number of primes, say P(x), less than a
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given integer x?” This question guides the rest of the study concerning the third major area of
prime numbers (Taylor 80-81).
The Prime Number Theorem: Distribution of Prime Numbers
The discovery and proof the Prime Number Theorem are two of the major achievements in
analytic number theory. Though the primes may go on forever, the fact that their frequency
decreases has been the focus of attention. A study initiated by Gauss in 1792 was aided by a
table published by Johann Lambert a few years before. The table included primes less than
102,000. If π (x) denotes the number of positive primes not exceeding x then what Gauss did
was to consider how π (x) grows with x. He counted the primes in successive intervals of fixed
length, obtaining a table in which ∆( x) = {π ( x) − π ( x − 1000)}/ 1000 :
Table 4
π (x)
x
∆(x)
1000
168
.168
2000
303
.135
3000
430
.127
4000
550
.120
5000
669
.119
6000
783
.114
7000
900
.117
8000
1007
.107
9000
1117
.110
10000
1229
.112
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The frequency ( ∆(x ) ) of primes seemed to be slowly decreasing, so Gauss took the reciprocal of
∆(x) and compared it with various functions. For the natural logarithm, this gives the following:
Table 5
x
1000
2000
3000
4000
5000
6000
7000
8000
9000
10,000
∆ ( x)
.168
.135
.127
.120
.119
.114
.117
.107
.110
.112
1/ln x
.145
.132
.125
.121
.117
.115
.113
.111
.110
.109
The good match strongly supports the guess that ∆( x) is approximately 1/ln x (LeVeque
Fundamentals 4).
The Prime Number Theorem describes the asymptotic density of the primes and states that,
“As x increases through the sequence of the positive integers, the number of primes less than x
tends to become approximately equal to
density of primes in the vicinity of x is
x
” (Ogilvy 93). The approximate or asymptotic
ln x
1
for large x. In other words, the probability that a
ln x
number of about the size of x is prime is
1
(Ogilvy 94). As formally stated by Legendre, “The
ln x
number of primes less than a given value is asymptotically that value, divided by its logarithm.”
For example, the number of primes less than 1 million is about 72,000, and the number of primes
less than 1 billion is about 48 million. In standard mathematical notation, the Prime Number
Theorem is written as: “The number of primes less than N is approximately equal to N/ln(N).
Also, the Nth prime is approximately equal to N x log(N) (Rockmore 35-36). It is important to
note that the logarithm involved is the natural log, with base e. The theorem tells us about how
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densely scattered the primes are in any given range. The Prime Number Theorem says that “the
gaps between the successive primes increase in length with increasing x, and it approximately
quantifies the increase” (Ogilvy 94).
The first fifty million primes have been calculated, and the differences between
successive primes, as well as the frequencies of these differences were tabulated. For example,
the first occurrence of 99 consecutive composite numbers is between the two primes 396,733 and
396,833. The gaps do not average 99, however, until we are in the range where x has an
estimated 44 digits (Caldwell).
One aim of analytic number theory has been to refine the Prime Number Theorem to a
form that gives a close approximation of the exact number of primes less that a given x. A
famous estimate is denoted by Li(x) and is equal to the following:
x
du
∫ ln u
(please note that the natural log is used).
o
This estimate gives a close approximation in the region where we can get an actual count of the
number of primes. In Table 6, N is the exact number of primes less than x. The relative error
decreases rapidly as x increases, and all values of Li(x) are greater than the correct N. Also, in
the chart, d is always positive and on the increase (Ogilvy 96).
Table 6
π (x)
x
Li(x)
d= Li(x)- π (x)
d/N= rel. error
1,000
168
178
10
.060
10,000
1,229
1,246
17
.014
100,000
9,592
9,630
38
.004
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1,000,000
78,498
78,628
130
.0017
10,000,000
664,579
664,918
339
.0005
In 1914, British mathematician J. E. Littelwood proved that the sign of d changes infinitely
often if the chart continues indefinitely, for example, at numbers x0<x1<x2<…, where xn tends
toward infinity (Ribenboim 180). The answer to how large x must be before d becomes negative
for the first time is not known. However, S. Skewes derived an upper limit by proving that for
some x, which he guarantees to be less than S, the sign of d will have changed. S is called
Skewe’s Number and is defined as
e
S = ee
79
.
The magnitude of S is so large that is it useful in theory only (Ogilvy 96-97).
Where to go from here?
For centuries, studies have propelled the branch of prime numbers in Number Theory. The
questions surrounding prime numbers develop naturally from an early understanding of
definitions and theory, but true comprehension of the magnitude of prime number studies
requires advanced skills. Though proofs have been used to show that there is an infinite amount
of prime numbers, the never-ending existence of a “next” prime number is still a concept that
seems to baffle logical reasoning. Furthermore, the continual search for a direct way to determine
whether or not a natural number is prime is something that both beginner mathematicians and
experts in the field can enjoy. The Prime Number Theorem has a lasting place in mathematical
history, forever among the great theorems of its time.
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Today, the study of prime numbers has progressed rapidly, especially with an increased
reliance on the ability of computer technology. Worldwide, computers are connected to systems,
constantly searching for the next prime number. Furthermore, prime numbers are the basis of
many public-key cryptography codes, maintaining security systems for businesses and banks.
Therefore, the findings and progress made by mathematicians decades and centuries ago are
fundamental to present day research. Every question concerning prime numbers is essential to
understanding their important role both in society and also in the study of Number Theory.
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Works Cited
Boyer, Carl B., and Uta C. Merzbach. A History of Mathematics. New York: Wiley,
1991. Print.
Caldwell, Chris. K. “The First Fifty Million Primes.”The Prime Pages.
< http://primes.utm.edu/lists/small/millions/>.
Dummit, David Steven and Richard M. Foote. Abstract Algebra, 3rd Edition. 2003.
Du, Sautoy Marcus. The Music of the Primes: Searching to Solve the Greatest Mystery in
Mathematics. New York: Perennial, 2004. Print.
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