Problem A: [BoH79]
Let r be a positive rational number but not an integer.
Prove that there are infinitely many positive integers n
such that |_n*r_| is prime.
Problem B: [Man77]
Prove that the integer u_n = |_ n^2 / 3 _| is a prime for
only a finite number of positive integers n.
Problem C: [Mul78]
Prove that |_ pi^n _| is a prime for
only a finite number of positive integers n.
Problem D: [Tod58, p353], [Esc11]
Let a and n be positive integers. Then
(a) |_ (a + sqrt(a^2 - 1))^n _| = 1 (modulo 2),
(b) |_ (a + sqrt(a^2 + 1))^n _| = n+1 (modulo 2).
Problem E: (Sillke)
Let x0 be the largest root of x^3 - 5 x^2 + x + 1. Prove that
|_ x0^n _| = 0 (modulo 2) for all positive integers n.
Problem F: [Putnam 1983/A-5]
Prove or disprove that there exists a positive real number u such that
|_ u^n _| = n (modulo 2) for all positive integers n.
Problem G: Connell Sequence [Con59], [IaM99], [LaP93], [Ste98]
Solution B:
case n=3m: u(n) = u(3m) = 3m^2. trivial
case n=3m+1: u(n) = u(3m+1) = (3m+2)m. trivial
case n=3m-1: u(n) = u(3m-1) = (3m-2)m. trivial
Therefore u(n) is prime if and only if u(3)=3 and u(4)=5.
Solution E: (Sillke)
x^3 - 5 x^2 + x + 1 = (x - x0)*(x - x1)*(x - x2) with
x0 = 4.744826077681923 = [4,1,2,1,11,3,31,..]
x1 = 0.604068139818794 = [0,1,1,1,1,9,4,..]
x2 = -0.348894217500717 = -[0,2,1,6,2,9,45..],
Consider the sequence a[n] = x0^n + x1^n + x2^n.
Then a[0] = 3, a[1] = 5, a[2] = 23, and
a[n] = 5 a[n-1] - a[n-2] - a[n-3] for all n>=3.
This sequence is only odd numbers.
As 0 < x1^n + x2^n < 1 for all positive integers.
Therefore we get |_x0^n_| = a[n] - 1 for all positive integers.
The first powers of x0 are 4.745, 22.513, 106.822, 506.852, 2404.925,
11410.950, 54142.971, 256898.982, 1218940.989, 5783662.994.
Solution F: (Sillke)
We can determine such values which are root of a cubic polynomial.
We start with a sequence a[n] = x0^n + x1^n + x2^n of odd numbers.
Let x0, x1, x2 be the roots of a cubic polynomial x^3 - c1 x^2 + c2 x - c3
then a[0] = 3, a[1] = c1, a[2] = c1^2 - 2 c2, and
a[n] = c1 a[n-1] - c2 a[n-2] + c3 a[n-3] for all n>=3.
So the sequence will be odd if c1, c2, and c3 be odd
(or if c1 be odd and c2 and c3 be even). For our purpose
we need two small real roots at least one negative.
The desired properties has the choice c1 = 7, c2 = -3, c3 = -3.
x^3 - 7 x^2 - 3 x + 3 = (x - x0)*(x - x1)*(x - x2) with
x0 = 7.352528664298343 = [7,2,1,5,8,4,1,..], 147/20 < x0 < 125/17
x1 = -0.838904510185198 = -[0,1,5,4,1,4,1,..], 26/31 < -x1 < 21/25
x2 = 0.486375845886855 = [0,2,17,1,5,1,1,..], 17/35 < x2 < 18/37
As 0 < x1^n + x2^n < x1^2 + x2^2 = a[2] - x0^2 < 55 - (147/20)^2 = 391/400
for all even n >= 2. Further 0 > x1^n + x2^n > x1^n > x1 > -21/25 for all
odd n. Therefore we get |_x0^n_| is a[n] for n odd and a[n]-1 for n>=2 even.
The first powers of x0 are 7.353, 54.060, 397.475, 2922.449, 21487.388,
157986.638, 1161601.286, 8540706.752, 62795791.204, 461707854.827.
References:
BoB93: J. Borwein, P. Borwein;
On the generating function of the integer part: [n*a + c]
Journal of Number Theory 43 (1993) 293-318
BoH79: I. Borosh, D. Hensley;
American Mathematical Monthly 86 (1979) 223 problem by Brorosh, Hensley
American Mathematical Monthly 87 (1980) 406 solution
Con59: Ian Connell;
American Mathematical Monthly 66:8 (Oct. 1959) 724 problem E1382 by Connell
American Mathematical Monthly 67:4 (Apr. 1960) 380 solution E1382
Eng98: Arthur Engel;
Problem-solving strategies,
Problem Books in Mathematics. New York, NY: Springer. x, 403 p. (1998)
ISBN 0-387-98219-1/hbk
Zbl 887.00002
Chap 14.6 Integer Function
Esc11: E. B. Escott;
American Mathematical Monthly 18 (1911) 230 problem AL-363 by Escott
American Mathematical Monthly 19 (1912) 51 solution AL-363 by Escott, ...
IaM99: Douglas E. Iannucci, Donna Mills-Taylor;
On Generalizing the Connel Sequence,
Journal of Integer Sequences 2 (1999) 99.1.7
http://www.research.att.com/~njas/sequences/JIS/IANN/iann1.html
LaP93: A. Lakhtakia, C. Pickover;
The Connell Sequence,
Journal of Recreational Mathematics, 25:2 (1993) 90-93
Nyb02: M. A. Nyblom;
Some Curious Sequences Involving Floor and Ceiling Functions,
American Mathematical Monthly 109 (2002) 559-564
sequence: 1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,6,6,...
sequence: 1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,...
sequence: excluding multiples of 3
sequence: excluding mth powers
Man77: Philip Mana;
Problem 358: Almost Always Composite,
The Fibonacci Quarterly 15 (1977) 285 problem by Philip Mana
The Fibonacci Quarterly 16 (1978) 474 solution by Graham Lord
Mul78: A. A. Mullin;
American Mathematical Monthly 85 (1978) 389 problem by A. A. Mullin
ReS90: A. J. Dos Reis, D. M. Silberger;
Generating non-powers by formula,
Mathematics Magazine 63 (1990) 53-55
Ste98: Gary E. Stevens;
A Connell-like Sequence,
Journal of Integer Sequences 1 (1998) 98.1.4
http://www.research.att.com/~njas/sequences/JIS/stevens.html
Tod58: Todhunter;
Algebra, 1858.
--
http://www.mathematik.uni-bielefeld.de/~sillke/
mailto:Torsten.Sillke@uni-bielefeld.de