ABC Conjecture Implies Infinitely Many non-x-fibonacci-wieferich Primes A Linear Case of Dynamical Systems

Transcription

1 ABC Conjecture Implies Infinitely Many non-x-fibonacci-wieferich Primes A Linear Case of Dynamical Systems Jun-Wen Wayne Peng Department of Mathematics, University of Rochester 3/21/2016 1/40

2 Motivation The Fibonacci sequence {F n } n=0, an ancient and beautiful integer sequence, is the following: 0, 1, 1, 2, 3, 5, 8, 13,.... This sequence is originally defined by the following recursive relation, with initial values F 0 = 0 and F 1 = 1, F n = F n 1 + F n 2 n 2. Let α > 0 and ᾱ be the roots of x 2 x 1 = 0. Then, the closed form of F n is given by F n = αn ᾱ n 5. 2/40

3 Motivation It will be very interesting to consider this sequence modulo m for some integer m. Let m be 11, and then the sequence {F n mod 11} is 0, 1, 1, 2, 3, 5, 8, 2, 10, 1, 0, 1, 1,.... Let m = 7, the sequence is 0, 1, 1, 2, 3, 5, 1, 6, 0, 6, 6, 5, 4, 2, 6, 1, 0, 1, 1,.... We should note that, in the both cases, the sequences repeat 0, 1, and so form a period. In fact, we can define a function π(m) := the period of the Fibonacci sequence modulo m. In 1960, D.D. Wall gave a paper in AMM(The American Mathematical Monthly), and studied the properties of the function π in that amusing paper. 3/40

4 Motivation Conjecture (wall s conjecture) π(p) π(p 2 ) p. Definition We call a prime p Fibonacci-Wieferich prime if π(p 2 ) = π(p). 4/40

5 Motivation We can treat more generally. Given a degree d linearly recurrence sequence X = {x m } with closed form x m = b 1 a m 1 + b 2 a m b d a m d for a i, b i K \ {0} where K is a number field, we can define, for some prime ideal p of K, π X (p e ) := period of {x m mod p e } where e is an integer, and X is not degenerated modulo p, i.e. a i, b i 0 mod p, and we then ask whether π X (p) = π X (p 2 ) or not. We will call these primes p X -Fibonacci-Wieferich primes. 5/40

6 ABC Conjecture Implies Infinitely Many non-x-fibonacci-wieferich Primes A Linear Case of Dynamical Systems Jun-Wen Wayne Peng Department of Mathematics, University of Rochester 3/21/2016 6/40

7 Wieferich Prime A Wieferich Primes is a prime satisfying the following equation 2 p 1 1 mod p 2. It has been conjectured that there are infinity Wieferich Primes although only two have been found so far. These two Wieferich primes are 1093 and In stead of proving Wieferich primes infinitely many, we can maybe show that the non-wieferich primes are finite. Unfortunately, in 1988, Silverman proved that there are infinitely many non-wieferich primes assuming ABC conjecture, so it is still uncertain that the number of Wieferich primes is finite or infinite. 7/40

8 Wieferich Prime What is the relation between Wieferich primes and Wall s conjecture? 8/40

9 Connection We first give a degree d linearly recurrence sequence X, and then give a prime ideal p of K where X is not degenerated modulo p. Then, X mod p e is period of length π if and only if b 1 a1 π + + b dad π x 0 mod p b 1 a1 π b d a π+1 d x 1 mod p holds.. b 1 a1 π+d b d a π+d 1 d x d 1 mod p 9/40

10 Connection If we consider each ai π to be a variable X i, then the above system of equations can be expressed as b 1 b 2 b d X 1 x 0 b 1 a 1 b 2 a 2 b d a d X x 1 mod pe. b 1 a1 d 1 b 2 a2 d 1 b d a d 1 d X d x d The matrix is a generalized Vandermonde matrix, so the the system has a unique solution which we should note that [1, 1,, 1] T is a solution. 10/40

11 Connection Therefore, we have a π 1 1 a π mod pe, 1 a π d and it implies that π = π X (p e ) = lcm{ord p e (a 1 ), ord p e (a 2 ),..., ord p e (a d )} 11/40

12 Observation π X (p) = lcm{ord p (a 1 ), ord p (a 2 ),..., ord p (a d )} π X (p) p f 1 ord p (a i ) ord p 2(a i ) ord p 2(a i ) = p ord p (a i ) If X = {a 1, a 2 } and a 1 a 2 = ±1 (multiplicative dependent), then an X -Fibonacci-Wieferich prime is just an a 1 -Wieferich prime. Otherwise, we only get p is an a i -Wieferich prime p is an X -Fibonacci-Wieferich prime. 12/40

13 An Easy but Useful Lemma Definition Let SQFP(I ) be the square-free part of an ideal I K, i.e. SQFP(I ) = p N p. Lemma (P.) If p is a prime dividing SQFP(a n i 1) for some n, then p is not an X -Fibonacci-Wieferich prime. So, if we can show that SQFP(α n 1) goes to infinity as n goes to infinity, we prove that there are infinitely many x-fibonacci-wieferich primes. 13/40

14 A Brief History Mathematicians were interested in Wieferich primes, because it was proved that if p is not a Wieferich prime and is the first case of Fermat s last theorem, then Fermat s last theorem will hold on this prime p. Two Chinese mathematicians, Zhihong Sun and Zhiwei Sun (they are brothers), show that the Fibonacci-Wieferich primes have the same property in 1992, so we sometimes call Wieferich-Fibonacci primes Wall-Sun-Sun primes, which are referred to the potential counterexamples of Fermat s last theorem. However, Fermat s last theorem was proved a year later in /40

15 abc conjecture We need several definitions for stating the ABC conjecture. Definition (Places of K) Let K be a number field, and M K be all places of K. A finite place v M K, correspondent to a prime ideal P, is defined as x v = N K/Q (P) v(x) x K. An infinite place v M K, correspondent to an embedding σ : K C, is defined as x v = σ(x) e x K where e = 1 if σ is totally real, otherwise, we let e = 2. 15/40

16 abc conjecture Definition (Logarithm Height) A logarithm height function h on P N (K) is, for all [x 0 :... : x N ] P N (K), h([x 0 :... : x N ]) = 1 [K : Q] v M K log max{ x 0 v, x 1 v,..., x N v } Definition (set of primes of a point) A set of prime ideals of a point p = [x 0 :... : x N ] P N (K) is I ([x 0 :... : x N ]) = {P O K v P (x i ) P(x j ) for some i, j} 16/40

17 ABC conjecture Definition (radical of a point) Radical of a point p P N (K) is rad(p) = P I (p) where we define N := log N K/Q /[K : Q]. Conjecture (ABC) N(P). Let p P 2 (K) be a point on the hyperplane X + Y + Z = 0. For any given ε > 0, we can find a constant C ε such that h(p) < (1 + ε) rad(p) + C ε ( ε ). 17/40

18 Our Result We call a field K ABC-field if ABC conjecture is true over K. Theorem (p.) There are infinitely many non-wieferich-fibonacci primes assuming K to be an ABC-field. For demonstrate our ideal, we can simply consider ABC conjecture over Q. 18/40

19 Skech of Proof Let c n = a n + b n where a, b Z with gcd(a, b) = 1. Claim: U n = SQFP(c n ) as n (Write c n = U n V n ) Suppose NOT! Say U n < C for some C for all n. Apply ABC, ε > 0, we have h( a n, b n, c n ) ε (1 + ε) rad(a n, b n, c n ). (1) We have and log U n + log V n h(a n, b n, c n ) (2) rad(a n, b n, c n ) < log U n log V n + log ab. (3) (1)+(2)+(3) ε (log U n + log V n ) a,b,ε log U n log V n 19/40

20 Sketch of Proof Then, since we assume U n is bounded a contradiction when ε < 1. Remark ε log V 1 n ε,a,b,c 2 log V n, This theorem tells that v n cannot too big (,i.e. has a upper bound). 20/40

21 Another Result of Applying n-term ABC Conjecture If we consider the canonical trace Tr : K Q given by Tr(k) = σ(k) k K, then Theorem (P.) σ:k C Let α be an algebraic integer with degree greater than 1. If the n-term ABC conjecture is true over the splitting field of α, then Tr(α m )s are not perfect power for all m large enough. 21/40

22 Lower Bound We define W α (B) := {p N(p) log B, and p is an α-fibonacci-wieferich prime.}, and we are going to show that there exists some constant C α such that W α (B) C α log B for all B greater enough. 22/40

23 Idea We hope p (γ m 1) where m is the first integer such that p (γ m 1), and it is well-known that γ m 1 = d m Φ d (γ). This goal can be achieve by showing that p Φ m (γ), i.e. p SQFP φ m (γ) =: U n, and showing p Φ d (γ) for any other d < m. This means γ is a root of Φ m (x) mod p, but it is not a root of Φ d (x) mod p. Then, we note that if we suppose p and m are coprime and γ is not a bed element modulo p, then x m 1 mod p is separable, i.e. when γ is a root of some factor Φ d (x) with d m, it won t be a root of others. Therefore, our first goal can be make by assuming that p SQFP(Φ m (γ)), γ is not a pole or zero divisor modulo p and p and n are coprime. 23/40

24 Idea We should also expect that the prime that we give in the latest slide is not a γ-fibonacci-wieferich prime These give me the first lemma. Lemma Let O K be the ring of integer of K, write (γ) = IJ 1 with I + J = O K. If we assume (n)ij + p = O K and p SQFP(Φ n (γ)), then we have m p = n and γ N K/Q(p) 1 1 mod p 2. 24/40

25 Idea Now, if we let N(SQFP(Φ n (γ))) > log(n)n(ij), (4) then we automatically get a prime p satisfying p + nij = O K and p SQFP(Φ n (γ)). We want to count how many n do we have for having (4), so we define {n N(SQFP(Φ n (γ))) > log(n)n(ij)}. We then find that we have a one-to-one map {n N(SQFP(Φ n (γ))) > log(n)n(ij)} M γ 25/40

26 Idea Then, it remains to figure out the constant B for which W γ (B) {n B N(SQFP(Φ n (γ))) > log(n)n(ij)}, and it gives the second lemma. Lemma W γ (B) {n (log B log 2)/h(γ) N(U n ) > log(n)n(ij)}. For simplify the notation, we define the following notations: (γ n 1) = u n v n w 1 n where uv + w = O K and u = SQFP(u n v n ) (Φ n (γ)) = U n V n W 1 n where UV + W = O K and U = SQFP(U n V n ) 26/40

27 Idea N(U n v n ) N(U n V n ) N(Φ n (γ)) N(v n ) N(U n ) ϕ(n) D 1 (0) log γ z dz Bounded above by ABC What we want cϕ(n) We get a lower bound, which is cϕ(n) nεh(γ) C ε,γ, for the N(U n ) 27/40

28 Idea We then let this lower bound greater than log(n)n(ij), which equivalently is cϕ(n) nεh(γ) C ε,γ log(n)n(ij) 0 (5) A theorem from analytic number theory comes to our rescue Lemma {n < Y ϕ(n) δn} ( 6 δ)y + O(log Y ) π2 Thus, for arbitrary δ > 0, if we choose ε = cδ/(2h(γ)) and let ϕ(n) δn, then cϕ(n) nεh(γ) cδn 1 2 cδn = 1 2 cδn > 0 28/40

29 Main Theorem Theorem For any non zero algebraic number γ X \ D 1 (0) where D 1 (0) is the complex unit circle with center at origin, there is a constant C γ such that we have assuming K is an abc-field. W γ (B) C γ log B B 0 29/40

30 Heuristic Result: Algebraic version Why do we conjecture infinitely many Wieferich primes? Even if we only find two Wieferich primes. 30/40

31 Let p be a prime integer. Considering 2 p 2 + kp mod p 2 for k {0, 1,..., p 1}, we suppose k is uniformly distributed on the set {0, 1,..., p 1}, which means that the possibility of getting k = 0 is 1/p. Thus, we expect to have 1 log log X p p X many Wieferich primes under X. 31/40

32 Heuristic Result: Algebraic version We use the similar idea to compute the number of {2, 3}-Fibonacci-Wieferich primes and should have p 1 p More generally, if we assume a 1,..., a d are multiplicatively independent, then the number of X -Fibonacci-Wieferich primes is p 1 p d. 32/40

33 Dynamical System Arithmetic dynamics is refer to the study of the number-theoretic properties of the iteration of polynomials or rational functions. Let f : X X be an endomorphism. We denote the following: 1 f n = n times {}}{ f f f 2 A point p X is periodic if f n (p) = p for some n > 1. 3 The orbit of p is the set O f (p) := {p, f (p), f 2 (p), f 3 (p),...} 4 Let f O K [x], and let O K be an ideal of K. We denote p := p mod I for p O K. Then, we define O f (p; I ) := {p, f (p), f 2 (p), f 3 (p),...} 33/40

34 Heuristic: Geometric Version Thus, via the terminology of arithmetic dynamics, a linear recurrence sequence {x n }, which has a recurrence relation x m+d+1 = c 1 x m+1 + c 2 x m+2 + c d x m+d with initial values x 1, x 2,..., x d, can be expressed by given the function f : R d R d with f : (x m+1, x m+2,..., x m+d ) (x m+2, x m+3,..., x m+d+1 ) 34/40

35 Heuristic: Geometric Version More generally, we can consider an algebraic variety X over the constant field K, an endormorphism f of X and a point on X. We should ask: Do we have infinitely many primes p such that O f (p, p) = O f (p, p 2 )? (6) 35/40

36 Heuristic: Geometric Version We propose the following conjecture. Conjecture 1 We have infinitely many primes p such that (6) holds if dim O f (p) = 1; 2 we have finitely many primes p such that (6) holds if dim O f (p) > 1. 36/40

37 Data: 2 n 1 37/40

38 Data: 2 n + 3 n 38/40

39 Data: the Fibonacci Sequence 39/40

40 Thank you! 40/40