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As an abstract artist inspired by a great love for the work of renaissance artist and mathematician Piero Della Francesca, I am now exploring the visual possibilities of working with the prime numb...

The modular multilevel converter (MMC) is attractive for high-power applications because of the advantages of its high modularity and high power quality. This paper proposes a prime factorization-b...

We describe a “handy” method, due to John Conway, for quickly finding all relatively small prime factors of 3-digit and 4-digit numbers.

Understanding the solution of a problem may require the reader to have background knowledge on the subject. For instance, finding an integer which, when divided by a nonzero integer leaves a remainder; but when divided by another nonzero integer may leave a different remainder. To find a smallest positive integer or a set of integers following the given conditions, one may need to understand the concept of modulo arithmetic in number theory. The Chinese Remainder Theorem is a known method to solve these types of problems using modulo arithmetic. In this paper, an efficient remainder rule has been proposed based on basic mathematical concepts. These core concepts are as follows: basic remainder rules of divisions, linear equation in slope intercept form, arithmetic progression and the use of a graphing calculator. These are easily understood by students who have taken prealgebra or intermediate algebra.

A ring R is strongly nil clean if every element in R is the sum of an idempotent and a nilpotent that commutes. We prove, in this article, that a ring R is strongly nil clean if and only if for any a ∈ R, there exists an idempotent e ∈ ℤ[a] such that a − e ∈ N(R), if and only if R is periodic and R∕J(R) is Boolean, if and only if each prime factor ring of R is strongly nil clean. Further, we prove that R is strongly nil clean if and only if for all a ∈ R, there exist n ∈ ℕ,k ≥ 0 (depending on a) such that an−an+2k∈N(R), if and only if for fixed m,n ∈ ℕ, a−a2n+2m(n+1)∈N(R) for all a ∈ R. These also extend known theorems, e.g, [5, Theorem 3.21], [6, Theorem 3], [7, Theorem 2.7] and [12, Theorem 2].

The M-graded domains , which are almost Schreier are classified under the assumption that the integral closure of R is a root extension of R, where M is a torsion-free, commutative, cancellative monoid. In the case that D[M] is a commutative monoid domain it is shown that if M conical and is a root extension, then D[M] is almost Schreier if and only if M and D are almost Schreier. If R=ℤ[nω] is an order in a quadratic extension field of ℚ, it is shown that the conditions; R[X] is IDPF; R[X] is inside factorial; R[X] is almost Schreier; is a root extension; and every prime divisor of n also divides the discriminant of the extension K/ℚ; are equivalent conditions.

The integer split quaternions form a noncommutative algebra over ℤ. We describe the prime and maximal spectrum of the integer split quaternions and investigate integer-valued polynomials over this ring. We prove that the set of such polynomials forms a ring, and proceed to study its prime and maximal ideals. In particular we completely classify the primes above 0, we obtain partial characterizations of primes above odd prime integers, and we give sufficient conditions for building maximal ideals above 2.

We present an integer rank reduction formula for transforming the rows and columns of an integer matrix A. By repeatedly applying the formula to reduce rank, an extended integer rank reducing process is derived. The process provides a general finite iterative approach for constructing factorizations of A and A T under a common framework of a general decomposition V T AP = Ω. Then, we develop the integer Wedderburn rank reduction formula and its integer biconjugation process. Both the integer biconjugation process associated with the Wedderburn rank reduction process and the scaled extended integer Abaffy–Broyden–Spedicato (ABS) class of algorithms are shown to be in the integer rank reducing process. We also show that the integer biconjugation process can be derived from the scaled integer ABS class of algorithms applied to A or A T . Finally, we show that the integer biconjuagation process is a special case of our proposed ABS class of algorithms for computing the Smith normal form.

It is proved that an integer n?=?2 is a prime (resp., composite) number if and only if there exists exactly one (resp., more than one) nth-degree monic polynomial fwith coefficients in Zn, the ring of integers modulo n, such that each element of Znis a root of f. This classroom note could find use in any introductory course on abstract algebra or elementary number theory.

An efficient computation of scalar multiplication in elliptic curve cryptography can be achieved by reducing the original problem into a chain of additions and doublings. Finding the shortest addition chain is an NP-problem. To produce the nearest possible shortest chain, various methods were introduced and most of them depends on the representation of a positive integer n into a binary form. Our method works out the given n by twice decomposition, first into its prime powers and second, for each prime into a series of 2's from which a set of rules based on addition and doubling is defined. Since prime factorization is computationally a hard problem, this method is only suitable for smooth integers. As an alternative, the need to decompose n can be avoided by choosing n of the form [image omitted] . This shall not compromise the security of ECC since its does not depend on prime factorization problem. The result shows a significant improvement over existing methods especially when n grows very large.

Let A and B be finite groups and let S be the set of all extensions of A by B. A group G is called an extension cover of (A, B), if G contains all extensions in S as subgroups of G. A group G is called a minimal extension cover if G is an extension cover of minimal order. Let be the prime factorization of the odd number n and define . The group D n 1 ×…×D n k × Z 2 is the unique minimal extension cover of (Z n , Z 2). This article also constructs a minimal extension cover of (Z 2 n , Z 2). Some conjectures about minimal extension covers are examined as well.

Classifying the positive integers as primes, composites, and the unit, is so familiar that it seems inevitable. However, other classifications can bring interesting relationships to our attention. In that spirit, let us classify positive integers by the number o? principal divisors they possess, where we define a principal divisor of a positive integer n to be any prime-power divisor pa \ n which is maximal (so p is prime, a is a positive integer, and pa+l is not a divisor of n). The standard notation pa \\n can be read as "/?fl is a principal divisor of n." The Fundamental Theorem of Arithmetic is usually stated in a form emphasizing how primes enter the structure of the positive integers, such as: Every positive integer is the product of a unique finite multiset of primes. (Recall that a multiset is a collection of elements in which multiple occurrences are permitted.) Alternatively, the Fundamental Theorem of Arithmetic can be stated in a form that focuses on how maximal prime powers enter the structure of the positive integers, such as: Every positive integer is the product of a unique finite set of powers of distinct primes. Consequently every positive integer is the product of its principal divisors, and every finite set of powers of distinct primes is the set of principal divisors of a unique positive integer. Of course, the number of principal divisors of n is equal to the number of distinct prime factors of n, but here the principal divisors are the simple structural components of n, whereas the distinct prime factors are but a shadow of that structure. Readers who find the present paper of interest might find similar interest in [6], where upper bounds on the sum of principal divisors of n are established by elementary means. For each integer n > 0, let Pn be the set of all positive integers with exactly n principal divisors, so Pq = {1}, and

of polynomials R[x] over R, namely the ring ^[[x]] of formal powers series in one variable over R, is hardly ever mentioned in such a course. In most cases, it is relegated to the homework problems (or to the exercises in the textbooks), and one learns that, like R[x], R[[x]] is an integral domain provided that R is an integral domain. More surprising is to learn that, in contrast to the situation of polynomials, in R[[x]] there are many invertible elements: while the only units in R[x] are the units of R, a necessary and sufficient condition for a power series to be invertible is that its constant term be invertible in R. This fact makes the study of arithmetic in /?[[*]] simple when R is a field: the only prime element is the variable x. As might be expected, the study of prime factorization in Z[[x]] is much more interesting (and complicated), but to the best of our knowledge it is not treated in detail in the available literature. After some basic considerations, it is apparent that the question of deciding whether or not an integral power series is prime is a difficult one, and it seems worthwhile to develop criteria to determine irreducibility in Z[[x]] similar to Eisenstein's criterion for polynomials. In this note we propose an easy argument that provides us with an infinite class of irreducible power series over Z. As in the case of Eisenstein's criterion in 7L\x\, our criteria give only sufficient conditions, and the question of whether or not a given power series is irreducible remains open in a vast array of cases, including quadratic polynomials. It is important to note that irreducibility in Z[x] and in Z[[x]] are, in general, un related. For instance, 6 + x + x2 is irreducible in Z[x] but can be factored in Z[[x]], while 2 + Ix + 3x2 is irreducible in Z[[x]] but equals (2 + x)(l +3x) as a polyno mial (observe that this is not a proper factorization in Z[[x]] since 1 + 3x is invert

We study the connection between products of Sylow subgroups of a finite group G and the solvable residual of G. Let Π(𝒫) be a product of Sylow subgroups of G such that each prime divisor of |G| is represented exactly once in Π(𝒫). We prove that there exists a unique normal subgroup N of G which is minimal subject to the requirement Π(𝒫) N = G. Furthermore, N is perfect, and the product of all of these subgroups is the solvable residual of G. We also prove that the solvable residual of G is generated by all elements which arise from non-trivial factorizations of 1 G in such products of Sylow subgroups.

Given a polynomial f (x), we study the possibility of expressing it as the composition of two non-constant and non-linear polynomials. In this case f(x) is said to be composite otherwise it is prime. We give sufficient conditions for a polynomial to be prime in terms of its critical values and critical points. Given two polynomials, f (x) and h(x) we give methods to decide if h(x) is a right composition factor of f (x) and in that case to find the polynomial g(x) such that f = g ○ h. Finally we propose an algorithm to decompose a polynomial f (x) into its prime factors if one knows its list of critical points with their valencies.

Let G be a finite p-solvable group for a fixed prime p. Attach to G a graph Γ p (G) whose vertices are the non-central p-regular conjugacy classes of G and connect two vertices by an edge if their cardinalities have a common prime divisor. In this note we study the structure and arithmetical properties of the p-regular class sizes in p-solvable groups G having Γ p (G) disconnected.

The fundamental theorem of arithmetic is one of those topics in mathematics that somehow ‘falls through the cracks’ in a student's education. When asked to state this theorem, those few students who are willing to give it a try (most have no idea of its content) will say something like ‘every natural number can be broken down into a product of primes’. The fact that this breakdown always results in the same primes is viewed as ‘obvious’. The purpose of this paper is to illustrate with a number of examples that the ‘Unique Factorization Property’ is a rare property and the fact that the natural numbers possess this property is ‘fundamental’ to our understanding of this number system.

Elementary school students wrestling with decimals quickly realize that not all fractions are created equal. While the relatively awkward fraction 17/32 turns into the modestly nice finite decimal 0.53125, other seemingly simple ones, like 2/3 = 0.66666666... or 1/7 = 0.14285714..., confront us with the sophisticated notion of infinite repeating decimals. Can we find a finitely fair base, one in which all fractions have finite representations? It is natural to start our hunt for a finitely fair base by changing from the familiar base 10 to base b, where b is any integer greater than 1. Unfortunately, the following review of representations base b reveals that for any b some fractions must have infinite repeating representations, while other fractions have finite representations. Recall that O.aa2a3 .. .b = alb a/b + a3/b2 = a3/b3 l n/bn, where the subscripted b indicates the base and an is an integer satisfying 0 < an < b 1. A fraction p/q in reduced form has a k-place representation in base b exactly when q divides bk but doesn't divide bk-l. For example, 32 = 25 divides 105 but not 104, so base 10 uses five decimal places for 17/32. If q has a prime factor not in b, the reduced fraction p/q has an infinite repeating representation base b. Since no fixed b has every prime factor, every base has some fractions with infinite repeating representations. Clearly, a finitely fair base requires something new, a mathematical "deconstruction" of the idea of a base. The postmodern term deconstruction describes something mathematicians have done for two centuries: probe a familiar concept more deeply to expose new interpretations and understandings. An initial deconstruction of base in the next section leads to a finitely fair base, called base {n!}. A further deconstruction in the middle section leads more generally to variable bases, which we use to find bases that "fit" a given real number with a specified representation. The final section critiques bases in yet another way, leading to competing measures for finitely fair bases. The Prime Number Theorem enables us to approximate one of these measures in terms of the other one and so resolve the competition between them. For ease we consider only representations of real numbers between 0 and 1, although the reader is invited to extend these ideas to the integer parts.

(2003). An Extension of the Results of Servais and Cramer on Odd Perfect and Odd Multiply Perfect Numbers. The American Mathematical Monthly: Vol. 110, No. 1, pp. 49-52.

This paper is devoted to some ring theoretic properties of the jordanian deformation of the algebra of regular functions on the matrices with coefficients in an algebraically closed field of characteristic zero, and of the associated factor algebra . We prove in particular that the prime spectrum of is the disjoint union of five components, each of which being homeomorphic to the spectrum of a commutative (possibly localised) polynomial ring. So we can give an explicit description of the prime spectrum of , and check that any prime factor of satisfies the Gelfand-Kirillov property. Then we study the automorphism groups of the algebras and and prove that they are generated by linear automorphisms and exponentials of locally nilpotent derivations.

Public-key cryptosystems base their security on well-known number-theoretic problems, such as factorisation of a given number n . Hence, prime number generation is an absolute requirement. Many prime number generation techniques have been proposed up-to-date, which differ mainly in terms of complexity, certainty and speed. Pocklington's theorem, if implemented, can guarantee the generation of a true prime. The proposed implementation exhibits low complexity at the expense of long execution time.

s unavailable at this time... Mathematics Subject Classification (2000): 11A25, 11P81 Keywords: partition, composition, relatively prime, arithmetic functions, related numbers, inversion formulas, elementary theory of partitions, inverse Quaestiones Mathematicaes 24 (4) 2001, 565–573

We investigate when semigroup algebras K[S] of submonoids S of torsion free polycyclic-by-finite groups G are Noetherian unique factorization rings in the sense of Chatters and Jordan, that is, every prime ideal contains a principal height one prime ideal. For the group algebra K[G] this problem was solved by Brown.

Let R be a commutative ringG a free abelian group of finite rank n and R#G the corresponding skew group ring. We fix an integer m; 0 ≤ m ≤ n and a free subgroup Gm of G of rank m. We prove that if R is noetherian, if specG (R#Gm ) is (R,G)-normally separated and if the Laurent polynomial ring is catenary, then the ring R#Gm is G-catenary. In the particular case where R is an affine algebra over an algebraically closed field, we prove that if R is G-locally finite, then the skew group ring R#Gm is universally catenary and universally G-catenary. Furthermore Tauvel's height formula is valid in all prime factors and G-prime factors of R#Gm.

Let D be a domain with quotient field K. The ring of integer-valued polynomials over D is Int(D) := { f E K[S];f( D) C D) . We describe the divisorial prime ideals of Int(D) when D is a domain of Krull-type and, in particular, when D is also a d-ring.

Using the notions of the rank and the degree of a prime p≠2 relative to a residue class g(modp), where g is a positive integer with (g,p)= 1, we study the systems of primitive roots. This study leads to the understanding of the behavior of the discrete periodical signals whose values are the digits of the inverses of integers expressed in an arithmetic system with base g.

Let R be a commutative ring with identity and let M be an R-module. We examine the situation where for each prime ideal ρof R the set of all ρ-prime submodules of M is finite. In case R is Noetherian and M is finitely generated, we prove that this condition is equivalent to there being a positive integer n such that for every prime ideal ρ of R, the number of ρ-prime submodules of Mis less than or equal to n. We further show that in this case, there is at most one ρ-prime submodule for all but finitely many prime ideals ρ of R.

The prime-factor DCT algorithm of two relatively prime factors decomposing the DCT matrix into skew-circular correlation (SCC) matrices and circular correlation (CC) matrices is generalized to many factors. As a result, the DCT output components are divided into several parts, each part is a sum of many SCC/CC matrix-vector products. The number of matrix-vector products is further reduced since some SCC/CC matrices can be represented by linear combination of other SCC/CC matrices with ± 1 coefficients. Thus the number of multiplications and additions is also reduced achieving less number of multiplications and additions compared with other algorithms. Besides each part of the output components being independent, parallel implementation of the proposed algorithm is feasible.

We consider the special case of the two functions coarsest partitioning problem, where one of the functions is a cyclic permutation and the other arbitrary. Here we present a parallel algorithm on a CRCW PRAM that solves the above partitioning problem in O(α(n)log(β(n))) time using O(n) processors, where n is the set cardinality, β(n) is the number of distinct prime factors of n, and α(n) is the sum of the exponents of the primes in the factorization of n. In almost all cases the algorithm runs in O(loglognlogloglogn) time with n processors.

In the 1870's, W.S. Jevons anticipated a key feature of the RSA algorithm for public key cryptography, namely that multiplication of integers is easy, but finding the prime factors of the product is hard. He presented a specific ten-digit number whose prime factorization, he believed, would forever remain unknown except to himself. In this paper, it is shown that Jevons' number could have been factored relatively easily, even in his own time.

Let G(q) be a finite simple group of Lie type over a finite field of order q and d(G(q)) the minimal degree of faithful projective complex representations of G(q). For the case G(q) is a classical group we deter-mine the number of projective complex characters of G(q) of degree d(G(q)). In several cases we also determine the projective complex characters of the second and the third lowest degrees. As a corollary of these results we deduce the classification of quasi-simple irreducible complex linear groups of degree at most 2r r a prime divisor of the group order.

A primary pseudoperfect number (PPN) is an integer $K > 1$ such that the reciprocals of $K$ and its prime factors sum to 1. PPNs arise in studying perfectly weighted graphs and singularities of algebraic surfaces, and are related to Sylvester's sequence, Giuga numbers, Zn\'am's problem, the inheritance problem, and Curtiss's bound on solutions of a unit fraction equation. Here we show $K \equiv 6 \pmod{6^2}$ if $6\mid K$, and uncover a remarkable $7$-term arithmetic progression of residues modulo $6^2\cdot8$ in the sequence of known PPNs. On that basis, we pose a conjecture which leads to a conditional proof of the new record lower bound $k>10^{3.99\times10^{20}}$ on any non-trivial solution to the Erd\H{o}s-Moser Diophantine equation $1^n + 2^n + \dotsb + k^n = (k+1)^n$., Comment: 7 pages, 1 table

(2004). On Prime Factors of An - 1. The American Mathematical Monthly: Vol. 111, No. 3, pp. 243-245.

The paper presents both arithmetic and time complexities involved in the implementation of circular convolution of real valued data using various fast Hartley transform (FHT) algorithms. It is observed that the prime factor FHT (PFFHT) algorithm involves less arithmetic complexity and requires significantly less computation time compared with the radix-2 and split radix FHT algorithms for the implementation of circular convolution. Apart from that, the PFFHT offers closer choices for convolution lengths unlike the radix-2 or split radix algorithms.

Prime numbers are the multiplicative building bricks of the number system. According to the fundamental theorem of arithmetic, every integer number larger than 1 is either a prime or the product of a unique set of primes. In what follows, by an integer we will understand a positive integer. In multiplicative number theory each integer is a word, more exactly a commutative juxtaposition of primes. In this coding process each prime is employed according to a rigid rule (the gap between the consecutive multiples of a prime p is just p) and the set of prime numbers is like an alphabet that is self-generating in order to make the resulting code nondegenerate. But how are the prime numbers themselves generated? Contemplating successive gaps between consecutive primes or the number of prime factors of consecutive integers, we can only notice an apparently chaotic behavior of the prime numbers leading us to believe that their distribution law must be very complicated. There are two different ways of looking at prime numbers: globally and algorithmically. From an algorithmic point of view the process of generating prime numbers is relatively clear. The prime-number sieve, attributed to the ancient Greek scholar Eratosthenes, was one of the first step-by-step methods invented for distinguishing primes from composites among the numbers up to some predetermined limit: Take the number 2, eliminate its multiples; the next prime is 3, eliminate its multiples; the next prime is 5, eliminate its multiples, etc. Today, checking whether or not an integer is a prime is one of the first computer programs learned in. any programming language. Eratosthenes' sieve simply tells us what to do, step-by-step, for selecting the primes in a given set of consecutive integers without revealing any regularity in the distribution of primes. Those unhappy with an algorithmic approach have tried several ways to approach a global understanding of the behavior of primes. Many papers have dealt with the asymptotic behavior of different functions depending on primes. There is a rich literature on the subject (see for instance [15], [17], [3], [2]) using very subtle mathematical techniques. To give only one example, let 7T(x) denote the number of primes not exceeding the positive real number x. According to the celebrated prime number theorem (PNT), we have 7T(x) = x/ln x, (x -> oo), which means that the ratio of the two functions, namely wr(x)/(x/ln x), converges to 1 as x grows without bound, proved independently by J. Hadamard [9] and C. J. de La Vallee Poussin [14] using tools involving functions of complex variables. PNT is a superb example of extracting asymptotic order from chaos.

There are three famous unsolved mathematical problems in number theory, namely the theory of partitions, Fermat's 'Last Theorem', and the prime number theorem. A geometrical method and an analytical method of predicting the distribution of primes and non-primes based on visual information obtainable from a matrix map of divisibles is described. Past investigations tend to concentrate on properties of the prime numbers. The author feels that much information could be gathered by studying the distribution of both prime and non-prime numbers using a matrix map. Both methods give accurate, deterministic mathematical models of the distribution of primes and non-primes globally. The only problem is that there is no end to the prime number series and thus the prime number theorem remains unsolved.

Let p be an odd prime and k

We prove in this paper a structure theorem for prime rings whose symmetric ring of quotients has nonzero socle. Then this result is applied to prime rings satisfying a generalized identity, and to prime rings having an alternate involution.

(1993). Gaussian Elimination in Integer Arithmetic: An Application of the L-U Factorization. The College Mathematics Journal: Vol. 24, No. 1, pp. 67-71.

A necessary and sufficient condition for a positive integer to be prime is explored in terms of its number-theoretic ramifications and its generalization to arithmetic progressions. A computer experiment is shown describing how the main results were conjectured. Several open problems are posed for readers.

(1991). Image Expansion in Integer Arithmetic. The College Mathematics Journal: Vol. 22, No. 5, pp. 429-435.

The purpose of this paper is to generalize the concept of semi prime ideals in Γ-rings. We use a general definition of a regularity F for Γ-rings to define and F- prime ideal. Relationships between F-semi prime ideals of a Γ-ring M and F-semi prime ideals of the operator rings R and L are discussed. D-regularity, f-regularity and λ-regularity for Γ-rings are introduced and studied against the background of the concept F-semi prime ideal. Finally, D-, λ- and f-regular Γ-rings are characterized.

In this paper, we study the Two Functions Set Partition Problem, which is defined as follows:Given a set S of n elements, functions f 1 and f 2 from S to S, and an initial partition B= (B 1 B 2,…,B s) of S, find the coarsest refinement E = (E 1,E 2,…E t) of B such that for each i i= 1,2, and j, 1 ≦ j≦t,f i (E j ) ⊆ E k for some k. For the special case when f 1consists of a single cycle, we present an 0(n β(n)) algorithm, where β(n) is the number of distinct prime factors of n. β(n) is loglog n+ o(loglog.n) for almost all n, and is Θ(log n/log log n) in the worst case. This algorithm represents an improvement over the previously known O(nlogn) algorithm.

LetΓ be a mulitiplicatively closed set of nonzero idcals of a Noetherian ring R and for an ideal I of R let IΓ=∪{I:G;G∈Γlcub; Then IΓ is an ideal in R and it is shown that if P is prime divisor of IΓ, then P is a prime divisor of IG:G′ for all G,G′ In $GgR;. Alao, for a given filtration is a filtration on R and when φ is Noetherian several necessary and sufficient conditions are given for φ and φΓ to give linearly equivalent ideal topologies on R .

It is well known that Euclid's argument can be adapted to prove the infinitude of primes of the form 4k − 1. We describe a simple proof that the sum of the reciprocals of all such primes diverges. More generally, if q is a positive integer and H is a proper subgroup of the units group (Z/qZ) × , we show that � p prime p mod qH 1 p =∞ . 1. INTRODUCTION. Perhaps no argument in number theory is more famous than Euclid's proof of the infinitude of primes. It is a traditional exercise in a first course in number theory to extend Euclid's argument to certain special arithmetic progressions. For example, take the progression 2, 5, 8, 11 ,... consisting of the positive integers of the form 3m − 1. Let p1 ,..., pk be any finite list of primes of the form 3m − 1, and consider the integer P := 3p1 ··· pk − 1. Now P factors into primes, all of which are either of the form 3m + 1o r 3m − 1. The product of numbers of the form 3m + 1i s again of the form 3m + 1, whereas P has the form 3m − 1. Thus, P must have some prime divisor of the form 3m − 1, and clearly this prime cannot be any of the pi .I t follows that we can extend the list p1 ,..., pk indefinitely; that is, there are infinitely many primes in the progression 3m − 1. Reasoning along similar lines, one obtains a proof of the following result, which is given as an exercise in D. A. Marcus's text on algebraic number theory (5, Exercise 6, p. 205). We write p mod q for the element of Z/qZ representing the reduction of p modulo q. Theorem A. Let q be a positive integer. Suppose that H is a proper subgroup of the unit group (Z/qZ) × . Then there are infinitely many primes p for which p mod qH. Our primary objective here is to showcase a proof, almost as simple, for the follow- ing strengthening of Theorem A. Theorem 1. Let q be a positive integer. Suppose that H is a proper subgroup of the unit group (Z/qZ) × . Then �

As the name of the paper implies, a converse of Fermat's Little Theorem (FLT) is stated and proved. FLT states the following: if p is any prime, and x any integer, then xp ≡ x (mod p). There is already a well-known converse of FLT, known as Lehmer's Theorem, which is as follows: if x is an integer coprime with m, such that xm −1 ≡ 1 (mod m), and if there exists no integer e

It is well known that no nonconstant polynomial with integer coefficients can take on only prime values. We isolate the property of the integers that accounts for this, and give several examples of integral domains for which there are polynomials that only take on unit or prime values. Throughout this note, we let R denote an arbitrary UFD (unique factorization domain) that is not a field. Of course, R must be infinite. As is well known, any nonconstant polynomial with integer coefficients cannot take on only prime values. We may ask what property of the integers Z accounts for this. Here is the answer.

Let q1 = 2. Supposing that we have defined qj for all 1  j  k, let qk+1 be a prime factor of 1 + Q k j=1 qj. As was shown by Euclid over two thousand years ago, q1,q2,q3,... is then an infinite sequence of distinct primes. The sequence {qi} is not unique, since there is flexibility in the choice of the prime qk+1 dividing 1 + Q k j=1 qj. Mullin suggested studying the two sequences formed by (1) always taking qk+1 as small as possible, and (2) always taking qk+1 as large as possible. For each of these sequences, he asked whether every prime eventually appears. Recently, Booker showed that the second sequence omits infinitely many primes. We give a completely elementary proof of Booker's result, suitable for presentation in a first course in number theory.