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Universität des Saarlandes
FR 6.1 Mathematik
Prof. Dr. R. Schulze-Pillot
Computational Number Theory
Problem Sheet 8
Due date: Thursday, 22. 01. 2015, before the lecture
Problem 1. Let prime numbers p1 , . . . , pk and positive integers a1 , . . . , ar be given, such that all
prime divisors of the aj are among the pi .
Q
a) Prove that for r > k there exists a subset I ⊆ {1, . . . , r} such that i∈I ai is a square.
(Hint: Use linear algebra over the field F2 with two elements.)
b) Give an algorithm (in pseudo-code) which computes all such squares (you may use a suitable
linear algebra subroutine).
Implementations are welcome!
(4P)
Problem 2.
a) For each of the following numbers N , compute the values of
N + 12 , N + 22 , N + 32 , N + 42 , . . .
until you find a value N + b2 that is a perfect square a2 . Then use the values of a and b to
factor N .
i) 53357
ii) 34571
iii) 25777
iv) 64213
b) For each of the listed values of N , k, and binit , factor N by making a list of values kN + b2 ,
starting at b = binit and incrementing b until kN + b2 is a perfect square. Then take greatest
common divisors.
N = 143041
k = 247
binit = 1
N = 1226987
k=3
binit = 36
N = 2510839
k = 21
binit = 90
(4P)
Problem 3. Compute the following values of ψ(X, B), the number of B-smooth numbers between
2 and X.
a) ψ(25, 3)
b) ψ(35, 5)
c) ψ(50, 7)
d) ψ(100, 5)
e) ψ(100, 7)
(4P)
Problem 4. An integer M is called B-power-smooth if every prime power pe dividing M satisfies
pe ≤ B. For example, 180 = 22 · 32 · 5 is 10-power-smooth, since the largest prime power dividing
180 is 9, which is smaller than 10.
a) Suppose that M is B-power-smooth. Prove that M is also B-smooth.
b) Suppose that M is B-smooth. Is it always true that M is also B-power-smooth? Either prove
that it is true or give an example for which it is not true.
c) The following is a list of 20 randomly chosen numbers between 1 and 1000. Which of these
numbers are 10-power-smooth? Which of them are 10-smooth?
{84, 141, 171, 208, 224, 318, 325, 366, 378, 390, 420, 440, 504, 530, 707, 726, 758, 765, 792, 817}
d) Prove that M is B-power-smooth if and only if M divides lcm(1, 2, . . . , B) (the least common
multiple of the numbers between 1 and B).
(4P)