ON THE SOLUTIONS OF CERTAIN CONGRUENCES SAHAR SIAVASHI Master of Science, The University of Tehran, 2012 A Thesis Submitted to the School of Graduate Studies of the University of Lethbridge in Partial Fulfillment of the Requirements for the Degree MASTER OF SCIENCE Department of Mathematics and Computer Science University of Lethbridge LETHBRIDGE, ALBERTA, CANADA c Sahar Siavashi, 2017 ON THE SOLUTIONS OF CERTAIN CONGRUENCES SAHAR SIAVASHI Date of Defense: March 29, 2017 Dr. Amir Akbary-Majdabadno Supervisor Professor Ph.D. Dr. Hadi Kharaghani Committee Member Professor Ph.D. Dr. Nathan Ng Committee Member Associate Professor Ph.D. Dr. Behnam Seyed-Mahmoud Committee Member Associate Professor Ph.D. Dr. Howard Cheng Chair, Thesis Examination Com- Associate Professor Ph.D. mittee Abstract We study the solutions of certain congruences in different rings. The congruences include ap−1 ≡ 1 (mod p2); for integer a > 1 and prime p with p - a, and aj(m) ≡ 1 (mod m2); for integer m with (a;m) = 1; where j is Euler’s totient function. The solutions of these congruences lead to Wieferich primes and Wieferich numbers. In another direction this thesis explores the extensions of these concepts to other number fields such as quadratic fields of class number one. We also study the solutions of the congruence gm − gn ≡ 0 (mod f m − f n); where m and n are two distinct natural numbers and f and g are two relatively prime poly- nomials with coefficients in the field of complex numbers. iii Acknowledgments First I would like to thank my supervisor, Prof. Amir Akbary, for his guidance and help throughout the research development and writing the thesis and for his academic support that helped me to grow as a scholar. Secondly I would like to thank my committee members Prof. Hadi Kharaghani, Prof. Nathan Ng and Prof. Behnam Seyed Mahmoud for their valuable comments and support. I would also like to thank Prof. Howard Cheng for serving as the chair of my defence committee. Also I would like to thank my office mates Arnab Bose and Forrest Francis for their comments and suggestions about writing the thesis and programming. iv Notation • Throughout this thesis, unless otherwise stated, p is a prime, a;m; and k are integers greater than 1, and x is a positive real number. • We use the conventional asymptotic notations of analytic number theory. For two real functions f and g, we write f (x) =O(g(x)) or f (x) g(x); if there exists a positive real constant C such that j f (x)j ≤ Cjg(x)j; for sufficiently large values of x: We write f (x) = Oa(g(x)) or f (x) a g(x) to denote the dependence of the constant C to the parameter a: We also write f (x) = o(g(x)) if f (x) lim = 0: x!¥ g(x) By f (x) ∼ g(x) as x ! ¥ we mean that f (x) lim = 1: x!¥ g(x) • By Wa(x) we denote the set of Wieferich primes in base a up to x: More precisely, p−1 2 Wa(x) = fp ≤ x ; a ≡ 1 (mod p )g: In the same way we denote the set of non-Wieferich primes up to x; in base a; by c c Wa (x): Also we denote by Wa;k(x) the set of non-Wieferich primes up to x; in base c a; in the arithmetic progressions p ≡ 1 (mod k): Moreover, by Wa and Wa we mean, respectively, the set of Wieferich primes in base a and the set of non-Wieferich primes v NOTATION in base a. a1 ak • We write rad(n) to denote the radical of an integer n = p1 ··· pk ; that is defined as rad(n) = p1 ··· pk: • The quality of a positive integer n is denoted by l(n); that is defined as logn l(n) = : lograd(n) • For an natural number n; the n-th cyclotomic polynomial, Fn(x); is defined as 2kpi Fn(x) = ∏ (x − e n ): 1≤k≤n gcd(k;n)=1 • The order of a mod p; denoted ordp(a); is the smallest positive integer k for which ak ≡ 1 (mod p): • By j(x) we denote the Euler-totient function. • We denote the Euler quotient for two relatively prime integers a and m by q(a;m); which is defined as aj(m) − 1 q(a;m) = : m For prime m; the Euler quotient is called the Fermat quotient. • We denote the set of Wieferich numbers up to x; in base a; by Na(x); which is defined as j(m) 2 Na(x) = fm ≤ x ; a ≡ 1 (mod m )g: vi NOTATION In the same fashion, we define the set of non-Wieferich numbers up to x; in base a; c c and denote it by Na(x): Moreover, Na and Na are the set of Wieferich numbers in base a and the set of non-Wieferich numbers in base a. • The largest power of p in an integer n is denoted by np(n): • A modified form of the Fermat quotient is denoted by q(a; p) and is 8 <> q(a; p) if p 6= 2 or p = 2 and a ≡ 1 (mod 4); q(a; p) = > a+1 : 2 if p = 2 and a ≡ 3 (mod 4): • The set Sa is the set of primes generated by primes in Wa: It is defined inductively as follows. Let 8 > (0) < Wa [ f2g if n2(q(a;2)) ≥ 1; Sa = :> Wa otherwise. For i ≥ 1; let (i) (i−1) Sa = fp ; pjq − 1 where q 2 Sa g: ¥ (i) Then, we define Sa = [i=0Sa : • An algebraic number field is denoted by K and its ring of integers by OK: • The norm of an ideal a is denoted by N(a) and is defined it as jOK=aj: • By hpi we mean the ideal generated by p: • The generalized Euler totient function for an ideal a is denoted by j(a) and is defined as 1 j(a) = N(a)∏ 1 − ; pja N(p) where p is a prime ideal divisor of a: vii NOTATION • We denote the set of K-Wieferich primes in base a with norm not exceeding x by Wa(K;x): It is defined as N(p)−1 2 Wa(K;x) = fp 2 OK ; N(p) ≤ x and a ≡ 1 (mod p )g: • By hK we denote the class number of a number field K: p • We denote a quadratic field by Q( m); where m is a square-free integer. p p • We write Gal(Q( m)=Q) for the Galois group of the extension Q( m)=Q: • By Q(i) we denote the Gaussian rational field which is Q(i) = fa + bi ; a;b 2 Qg: We denote its ring of integers by Z[i] and it is called the Gaussian integers. • We write C[x] to denote the ring of polynomials with the coefficients in C: • If f (x) = a∏(x − ai); i where a 2 C and ai’s are distinct numbers in C; then we define the radical of f by rad( f ) = ∏(x − ai): i • The degree of a polynomial in C[x] is denoted by deg( f ). • We write f (i)(x) to denote the i-th derivative of f at x: viii Contents Contents ix 1 Introduction and statement of results 1 1.1 Wieferich primes and Wieferich numbers . .1 1.2 Wieferich primes and Wieferich numbers in number fields . .9 1.3 An exponential congruence in C[x] ...................... 17 2 Wieferich primes and Wieferich numbers 20 2.1 An improvement of Graves-Murty lower bound . 20 2.2 An improvement . 26 2.3 The largest known Wieferich numbers . 35 2.4 Density of Wieferich numbers . 39 2.5 Density of non-Wieferich numbers . 41 3 K-Wieferich primes and numbers 44 3.1 Wieferich primes in a quadratic field . 44 3.2 Wieferich numbers in a quadratic field . 47 4 An exponential congruence in C[x] 59 4.1 A finiteness theorem in C[x] ......................... 59 4.2 An effective finiteness result . 65 5 Concluding Remark 71 Bibliography 73 A Tables 75 ix Chapter 1 Introduction and statement of results 1.1 Wieferich primes and Wieferich numbers The study of primes has fascinated humans from early times. The primes were exten- sively studied by ancient Greek mathematicians. The fundamental theorem of arithmetic that every integer greater than 1 can be written uniquely as the product of certain primes, is first stated in Euclid’s Elements [8]. Some important classes of primes include, primes in arithmetic progressions, Mersenne primes, Wilson primes, and twin primes. A sequence of primes that is of our interest in this thesis is the so-called Wieferich primes. An odd prime p is called a Wieferich prime (in base 2), if 2p−1 ≡ 1 (mod p2): These primes first were considered by Arthur Wieferich in 1909, while he was working on a proof of Fermat’s last theorem. Fermat in 1637, in the margin of his copy of Diophantus’s Arithmetica, stated that the equation an +bn = cn; for n > 2; has no integer solutions (a;b;c); with abc 6= 0: Despite claiming that he knew the proof, he did not provide it. This statement became famous as Fermat’s last theorem. Sophie Germain was one of the mathematicians who had worked on Fermat’s last theorem (see [7] for a historical account). She showed that Fermat’s last theorem can be divided into two cases and she proved the first case for p < 100 ( More pre- cisely, the first case is the statement taht the equation ap +bp = cp has no nontrivial solution (a;b;c) where p - abc).