Homework assignments
(MATH 3163-001, Spring 2026)
Instructor: Gábor Hetyei Last update: Tuesday, February 3, 2026

Disclaimer: The information below comes with no warranty. If, due to typographical error, there is a discrepancy between the exercises announced in class and the ones below, or this page is not completely up to date, the required homework consists of those exercises which were announced in class. Check for the time of last update above. If, by my mistake, a wrong exercise shows up below, I will allow you extra time to hand in the exercise that was announced in class. If, however, exercises are missing because this page is not up to date, it is your responsibility to contact me before the due date. (No extra time will be allowed in that case.) This page is up to date if the last update happened after the last class before the next due date.

The deadlines do not apply to the Bonus questions, which expire only once we solve them in class, or on April 21 at latest.
Notation: In the table below, 1.1/1a means exercise 1, part a, in section 1.1.

No. Date due: Problems:
3 Tue 2/10 at noon 2.1/16;   2.2/2,4,10,14c
2 Tue 2/3 at noon 1.3/6,8b,14;   2.1/4,5b,10.
Bonus problems: Prove that the square root of a positive integer is either an integer or it is irrational. (B03, 5 points)
1 Tue 1/27 at noon 1.1/8,11;  1.2/15c (also write the greatest common divisor of 1003 and 456 as
1003 m+ 456 n, see the file notes-0122.pdf on Canvas for help);   and the following
Board problems:
  1. Prove by induction that 12+22+...+n2=n(n+1)(2n+1)/6. holds for n ≥ 1.
  2. The sequence a0, a1, a2,... is given by the initial conditions a0=4 , a1=-1 and by the recurrence an+2= an+1+2 ·an.
    Prove by induction that an=2n+3 · (-1)n.
  3. Prove that a divides b if and only if -a divides b.
Bonus problems:
  1. If a=q0b+r0 b=q1r0+r1 and r0=q2r1+r2 then express r2 as an integer linear combination of ua+vb of a and b. (B01, 4 points)
  2. 1.2/33, without using unique prime factorization (B02, 5 points)