DIOPHANTINE PROBLEMS, POLYNOMIAL PROBLEMS, & PRIME PROBLEMS. positive ) integer Q how close are you guaranteed to get to the circle with rationals

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1 DIOPHANTINE PROBLEMS, POLYNOMIAL PROBLEMS, & PRIME PROBLEMS CHRIS PINNER. Diophantine Problems. Given an arbitrary circle in R 2 and( positive ) integer Q how close are you guaranteed to get to the circle with rationals r q, s q with q Q? Since there are infinitely many points on the circle presumably one can do better than 2 Qq, the bound obtained by picking a point on the circle and then approximating that using the box principle. Exercise. Prove the box principle bound 2 Qq. 2. R = {a + b 4 : a, b Z} is Euclidean though not Norm-Euclidean. Find an explicit euclidean function, i.e. find a function φ : R N {0} such that φ(x) = 0 iff x = 0 and for any a, b in R there exist q, r in R with a = bq + r, φ(r) < φ(b). Exercise. Prove that R = {a + b D : a, b Z} is Euclidean under the norm φ(a + b D) = (a + b D)(a b D) = a 2 Db 2 when D = 2 or 3. What about D = 6? 3. It is claimed that x 3 + y 3 + z 3 = n has a solution x, y, z in Z for all n not of the form 9k ± 4 (it has been proved that every number not of that form can be written as a sum of four integer cubes). This has been verified for all n 000 except for n = 33, 42, 74, 4, 56, 65, 38, 366, 390, 420, 564, 579, 627, 633, 732, 758, 789, 795, 894, 906, 92, 933, 948, 975. Find solutions for any of these. Exercise. Prove that any n of the form 9k ± 4 can not be written as a sum of 3 cubes. See KC2H-MSM/mathland/math04/matb00.htm for some tables and references. 4. Mike Bennett has proved that 3 2 p q > 4q 2.5 for all rational p/q. Obtain effective bounds of this type for other algebraic numbers α of degree d 3 (or for 3 2 with a different exponent), i.e. α p c(α, κ) q > q d κ Date: June 2, 2005.

2 2 CHRIS PINNER for all p/q with some κ > 0 and explicit constant c(α, k). Exercise. Prove Liouville s lower bound - namely that if α is a root of some irreducible polynomial with integer coefficients and degree d then there is a constant c(α) such that α p q > c(α) q d for all p/q. 5. It is claimed that every algebraic number α of degree 3 or more is not badly approximable. That is, for any A > 0 there are infinitely many p/q with α p/q < /Aq 2, or equivalently α has unbounded partial quotients. Find an example where this can be proved. Exercise. Prove that quadratics are badly approximable - in particular for any δ > 0 the inequality p D q < (2 D + δ)q 2 has only finitely many solutions 6. Find real numbers/series with interesting continued fraction expansions. Exercise. Examine known examples such as e, cotanh(/n), or h= ±2 2h, then experiment. 7. Find Laurent series with interesting continued fraction expansions e.g. expansions where all the partial quotients are linear, or all have large degree or regularly spaced large degree quotients, or are polynomials with integer coefficients or have unexpectedly nice denominators, or whose truncations yield regularly spaced convergents. Then try and explain why they have that particular property. Exercise. a) Examine known examples such as i= ( + ) for k = 3 or k 4 x kh even, or h= x f h where f n is the nth fibonacci number, then experiment. b) Prove that if the λ i grow fast enough, namely λ n > 2 n j= λ i, then the truncations j i= ( ± ) are all convergents to x λi i= ( ± ). x λi 8. (Serre s challenge) Find all the solutions to the Fermat like equation x 4 + y 4 = 7z 4, x, y, z in N. Which primes are the sum of two fourth powers? Which integers? Exercise. Solve the Diophantine equations, x, y, z in N; a) x 4 + y 4 = 7z 4, b) x 2 + y 2 = 7z 2, c) x 4 + y 4 = z For a real number α and each real number γ not of the form n + mα, n, m in Z, define the inhomogeneous approximation constant M(α, γ) = lim inf q qα γ q and the inhomogeneous Lagrange spectrum L(α) = {M(α, γ) : γ R, γ n + mα, n, m Z}. Suppose that α = r + s D, D N squarefree, r, s in Q is a quadratic.

3 REU NUMBER THEORY PROBLEMS 3 (a) If γ = u + v D, u, v rational give a straightforward proof that M(α, γ) = M(α, γ) where α = r s D and γ = u v D, are the algebraic conjugates. (b) Prove that L(α) is the closure of the set of M(α, γ) with γ = u + v D, u, v rational. (c) The Barnes Swinnerton-Dyer conjectures state that there is a gap between the largest and second largest values in L(α). This has been verified for α whose continued fraction expansion has period one or two or all even quotients and many sporadic examples. Prove this for other examples or classes. (d) It is conjectured that L(α) always contains a Hall s Ray [0, c(α)]. Find classes of α where you can prove this. Exercise. Prove that the M(α, γ) The continued fraction algorithm finds the best approximations with the usual measure of closeness. Is there an algorithm to find the p-adically best rational approximations? Exercise. Find out what being p-adically close means!. Is σ(n) = σ(n + ) infinitely often (where σ(n) denotes the sum of the positive divisors of n). Are there any k for which it can be proved that σ(n) = σ(n + k) infinitely often? This is true for all k if σ(n) is replaced by d(n) the number of positive divisors. Exercise. solutions. Experiment to find k which have many solutions or apparently no

4 4 CHRIS PINNER For a polynomial define the Mahler measure the height length P l 2. Polynomial Problems P (z) = n a i x i = a n i=0 M(P ) = a n = n i= a i, and norms n i= (x α i ) n max{, α i }, i= H(P ) = n a i 2, i=0 P = sup P (z), P 4 = z = ( and N(P ) the number of non-zero coefficients of P. 0 ) P (e 2πit ) 4 4 dt,. (Prouhet-Tarry-Escott Problem). Find distinct sets of integers α,..., α N and β,..., β N (repeats allowed within the sets but no α i = β j ) with α j + + αj N = βj + + βj N for j =,..., N. Find examples for N = or N 3. Find an N for which no solution exists. Find infinitely many inequivalent solutions for N = 0 or N = 2. Exercise. a) Show that a solution with j k is equivalent to the following: i) N i= N xαi i= has a factor (x ) k xβi ii) N i= (x α i) N i= (x β i) has degree N k. b)show that for any k there will be a solution j =,.., k for suitably large N. 2. (Erdos-Szekeres problem) Define E N = E N = min (x α ) (x α N ) l α,...,α N min (x α ) (x α N ) α,...,α N It is conjectured for any positive constant β that EN > c (β)n β and E N > c(β)n β. A weaker claim would be EN 2N + 2 for any N except,2,3,4,5,6,8 (this has been verified by Roy Maltby for N = 7, 9, 0); prove this for some N. Exercise. Prove that EN 2N. Compute the optimal sets α,..., α N for small N. Prove that the conjectures for E N and EN are equivalent. 3. Find a polynomial P with a root off the unit circle such that P 4 or higher power divides some polynomial F with all coefficients a i = 0, or. This is true for P = l(x) the Lehmer polynomial l(x) = x 0 +x 9 x 7 x 6 x 5 x 4 x 3 +x+ but the corresponding F for l(x) (or any l(x m )) is still unknown. Prove that arbitrarily large multiplicity factors are possible. Alternatively, prove that there is a bound on the maximum multiplicity - this would prove Lehmer s conjecture.

5 REU NUMBER THEORY PROBLEMS 5 Exercise. a) Prove that if P is a polynomial with M(P ) < 2 then there is a {0,, +} polynomial F with P as a factor. It is unknown if one can do this with M(F ) = M(P ). It is unknown if there is a constant c such that M(P ) < c implies that P is a factor of a {+, } polynomial. b) Prove that any cyclotomic polynomial can occur as a factor of a {0, +, } polynomials to arbitrarily high multiplicity. 4. Find a Lehmer s problem lower bound log M(F ) > c > 0 for some subclass of non-cyclotomic polynomials. Such bounds do exist, for example Smyth s bound log M(F ) log for non-reciprocal polynomials. Also Borwein- Dobrowolski-Mossinghoff s bound log M(F ) 8 log 5 for any polynomial with all odd coefficients and no cyclotomic factor. Can the assumption of no cyclotomic factors be dropped? Their class includes the {+, } polynomials but what about {0, +, } polynomials? Dobrowolski has proved a bound M(F ) + 39e2.27N(F )N(F ) Can such bounds (or one involving some dependence on H(F )) be obtained for the non-cyclotomic factors f of F rather than F itself? Exercise. a) A conjecture of Schinzel & Zassenhaus says that there is a constant c > 0 such that every non-cyclotomic polynomial of degree n has a root α with α > + c n. Prove that this would follow immediately from a Lehmer lower bound. b) Look up the proof of the above bounds or the current best general bound of ) 3. Dobrowolski log M(F ) > c ( log log n log n 5. A Lehmer Problem lower bound M(P ) is equivalent to the bound Ω (F ) log H(F ) log on the number of factors of F which are not x or cyclotomic (recall a cyclotomic polynomial is one whose roots are all roots of unity, ie a factor of x n for some n). In fact it is equivalent to the same bound on the maximum multiplicity m(f ) of such a factor. Obtain a bound on Ω (F ) which just depends on H(F ) however bad that dependence e.g. H(F ) H(F ) would do. It can be shown that the number of non-cyclotomic factors of an integer polynomial of degree n is bounded by C(ε)n ε N(F ) 2. Can the N(F ) 2 be replaced by N(F )? Could such a bound hold for Ω (F ). Exercise. a) Prove that a Lehmer bound M(P ) > C for all polynomials not consisting solely of cyclotomics or powers of x leads to a bound log H(F )/ log C for Ω (F ) and m(f ). Note that log M(F ) = log F (e 2πit ) dt log F (e 2πit ) 2 dt = log H(F ). 0 b) Show the bound m(f ) N(F ) H(F ). 6. For a positive integer n what is the largest k such that (x ) k is a factor of some polynomial of degree n with coefficients {0,, +} or coefficients {, +}? It can 0

6 6 CHRIS PINNER n be shown that c log n < max k < c 2 n for {0,, +} and c log n < max k < log c 2 n 2 log log n for {, +}. Exercise. a) Show that k i= (x2i ) is a {+, } polynomial and hence prove the lower bound for {, +} polynomials. b) Use the box principle to prove the lower bound for {0,, +} polynomials. 7. (Zeros of {+, } polynomials). Poonen has conjectured that as N the number of zeros of a reciprocal {+, } polynomial of degree 2N which lie on the unit circle z = tends to infinity. Compute examples with few zeros for small values of N. Do any patterns recur? Is c log log N a reasonable lower bound? Show that for sufficiently large N such polynomials have at least 2 pairs of zeros (if α is a root then so is its conjugate α ). Erdelyi has conjectured that there is a constant c > 0 such that any {+, } polynomial has at least one zero in the annulus c n < z < + c n. Exercise. a) Prove that any zero of a {+, } polynomial must satisfy 2 < α < 2. b) Show that a reciprocal {+, } polynomial has at least one zero on the unit circle. 8. (Integer Chebyshev Problem). For a real interval [α, β] define f [α,β] = j=0 sup f(z), z [α,β] n Z n = a j x j : a j Z, and the integer Chebyshev constant ( ) N C(α, β) = lim min f [α,β]. N f Z n Find C(α, β) for an interval of length < 4. Find good bounds for some intervals such as C(, ), C( 2, 2 ) or C(0, 2 ). Improve the inequality for the most interesting interval C(0, ) Find upper bounds C(0, m ) < m+δ m and hence lower bounds a n δ m n for the trace of (almost all) integer polynomials p n = z n + a n z n + + a 0 with all their roots real and positive. Can you do better than C(0, 00 ) < / or improve Smyth s result a n.77...n. It is conjectured (Schur-Siegel-Smyth trace problem) that for any δ > 0 there are only finitely many with a n < (2 δ)n. Exercise. Find a proof that n min c i R xn + c j x j = 2 n T n(x) = [,] 2 n. [,] j=0

7 REU NUMBER THEORY PROBLEMS 7 where T n (x) is the nth Chebyshev polynomial, T n (cos θ) = cos(nθ), and hence that n n lim min n c i R xn + c j x j = (β α). 4 j=0 [α,β] for the non-integer Chebyshev constant. b) Hence deduce that C( 2, 2) = achieved by the integer polynomials T n (x), and C(α, β) = for all β α 4 achieved by f =. 9. (Littlewood s Problem) Are there positive constants c and c 2 such that for any n there exists a {+, } polynomial p of degree n with c n + p(z) c2 n +, z =. Erdos conjectures that c 2 is bounded away from. Beck has shown the result if one allows the coefficients to be 400th roots of unity. Can one produce such bounds for other restrictive classes (or a simpler proof with a value worse than 400). Note one can take c = δ n, c 2 = + δ n with δ n 0 as n if one allows any a i =. Find families of polynomials like the Shapiro polynomials which at least satisfy an upper bound? What about the lower bound? Exercise. a) Prove that sup z = p(z) H(p) = n +. b) Prove that the Shapiro polynomials P 0 (z) =, Q 0 (z) =, P n+ (z) = P n (z) + z 2n Q n (z), Q n+ (z) = P n (z) z 2n Q n (z) are {+, } polynomials of degree 2 n satisfying P 2 n+ + Q 2 n+ = 2 n+2 on z = and hence satisfy a Littlewood type upper bound P n+ (z) 2 degree(p n+ ) (Merit Factors) It is conjectured that there is a constant c > 0 such that P 4 > ( + c) n + for all polynomials of degree n with coefficients +, Can one obtain the much weaker bound P 4 > ( ) n n+ for n 3, proving the non-existence of Barker polynomials for n > 2? Exercise. a) Observe that p(x)p(/x) = (n + ) + n j= n,j 0 c jx j opposite parity to n j and thus that with c j of 4 n p 4 = (n + ) 2 + c 2 j ( n + + n + δ ) 4 n + j= n,j 0 with δ = 0 when n is even and when n is odd. Barker polynomials have c j and hence equality in this bound. b) Compute examples with small 4-norm. Experiment with the shifted Fekete polynomials f t p(x) = p k=0 ( k+t p ) x k.. For a {+, } polynomial of degree n we have M(F ) H(F ) = n +. How close can one come to achieving this. Exercise. Experiment numerically and try to find classes of {+, } polynomals with large measure.

8 8 CHRIS PINNER 3. Prime Problems. The precision of the prime number theorem says that the nth prime p n satisfies p n = (+δ n )n log n with δ n 0 as n. Give a mathematically unsophisticated proof of the much weaker bound p n < n 2 for n 2 (or other polynomial bound). Exercise. Show that the Fermat numbers are pairwise coprime and hence deduce that p n 2 2n +. The bound p n < 4 n is also easy to obtain. 2. a) Mill s Theorem says that there is a real number θ with θ 3n prime for all n. Find a rational θ that produces primes for n =,.., 5 or more (as an example θ = works for n =,.., 4). It is conjectured that the θ in Mill s Theorem can be taken close to.3. b) Find a rational θ so that θ n produces primes for n =,.., 9. (for example θ = works for n =,.., 8). Is there a θ that works for all n? c) Are there infinitely many primes p, q, r with p = α, q = α 2, r = α 3? This has been proved for pairs p, q. Exercise. a) Show that p m = 0 2n α 0 2n 0 2n α where α = = p m 0 2m. m= b) Look at a proof of Mill s theorem. 3. Find prime producing polynomials. e.g. x 2 79x + 60 is prime for 0 x 79. Particularly interesting are polynomials connected with unique factorisation domains (e.g. if 2x 2 + 2tx ± is prime for x < t then Q( t 2 ± 2) is a UFD). Exercise. Show that a polynomial with integer coefficients must take infinitely many composite values. 4. Find a primality test for the nth Cullen number C n = n2 n +. Can one say anything about the prime factors of C n? Find large Cullen primes. Exercise. a) By considering C p prove that there are infinitely many composite Cullen numbers. b) Look at the Lucas-Lehmer and Pepin s test for the primality of Mersenne numbers M p or Fermat numbers F n. Prove that prime factors of M p or F n must be of the form 2kp + and 2 n+ k + respectively. 5. Compute R(p) the smallest primitive root mod p for many primes p. Is 2 a primitive root for % of the primes as predicted by the generalised Riemann hypothesis? Is the data consistent with the Hooley conjecture R(p) < c log p log log p or only the GRH bound R(p) < c log 6 p or does it suggest that the known lower bound, R(p) > c log p log log log p infinitely often, is best possible. Exercise. Prove that a is a primitive root mod p iff a (p )/q (mod p) for all primes q (p ).

9 REU NUMBER THEORY PROBLEMS 9 6. Compute N(p), the smallest non-square mod p, for many primes p. Investigate the occurrence of long strings of successive squares or non squares ( ) S(p) = max x A,B p. A x B Is the data consistent with the conjectures N(p) < c log p log log p or S(p) < c p log log p or even suggest optimal constants c? Exercise. Why is N(p) always a prime? Find congruence conditions that make N(p) = 2, 3, 5 or For a composite number n let S(n) = {a : a is a strong-pseudoprime base a} E(n) = {a : a is an Euler-pseudoprime base a} and let W (n) be the least witness i.e. the smallest element not in S(n). Show that there are sequences n i and n j with #S(n i ) lim = i n i 4, lim #E(n j ) = j n j 2. Find the smallest n with W (n) = 2. Compute W (n) for many n and compare this with the Extended Riemann hypothesis bound W (n) < 2 log 2 n. Find an n ±2 (mod 5) which is both a pseudoprime base 2 and Fibonacci pseudoprime, or prove that none exist ($620 has been offered either way). 8. Does the period of the decimal expansion of p many primes p? have period (p ) for infinitely Exercise. Prove that the period of /p divides (p ). Show that it equals (p ) exactly when 0 is a primitive root mod p. Can a composite n have period n? 9. Computational Problems: a) Are there consecutive primes p < q < r with p (qr + ) beyond 6 ( )? b) Determine whether 2 B(3), 2 B(6),2 B(27), are prime or composite where B(p) = (2 p + )/3 ($000 has been offered for each). c) Find a Mersenne number or Fermat number with a square factor. d) Show that the sequence c = 2, c i+ = 2 ci are not all prime. 0. a) Show that if x p = x + then x generates F p p. b) Show that the Galois group of x n + x + 3 is S n. References [] Peter Borwein, Computational Excursions in Analysis & Number Theory, Springer-Verlag, [2] Crandall & Pomerance, Prime Numbers, Springer, 200. [3] Richard K. Guy, Unsolved Problems in Number Theory, Springer (2nd ed. 994, 3rd ed. 2004?). Department of Mathematics, Kansas State University, Manhattan, KS address: pinner@math.ksu.edu