Number Theory - Project Suggestions

 

1.     Explore the sequences of quadratic residues modulo m for numbers of the form m=pn for p a fixed prime. Can you find any patterns to the number of quadratic residues or the distribution of the residues themselves?

2.     Explore the sequences of quadratic residues modulo m for numbers of the form m=p2 as p moves through the odd primes. Can you find any patterns to the number of quadratic residues or the distribution of the residues themselves?

3.     Define a cubic residue modulo m. Generate sets of  cubic residues mod p for p a prime and discuss any patterns you find.

4.     Update problem #44 on page 33, concerning Mersenne Primes.  The information in the problem is a few years out of date.  Find a test for primality for 2n-1 in the literature and apply it to some large cases.

5.     Mersenne primes are primes of the form 2n-1.  We saw in the second Maple exercise that n had to be a prime if  2n-1 was to be a prime.  Are there any pseudoprimes (to some base) in the set of numbers {2n-1} where n is a prime?

6.     Look for primes in a sequence like 3n-2 or 3n +2 or 5n +2 etc. Are there necessary or sufficient conditions on n?

7.     What is the most up to date information on perfect numbers? What is the largest one known now? Are there any odd perfect numbers?

8.     Fermat’s Theorem says that if p is prime and (a,p)=1, then ap-1º1 (mod p).  What happens if p is of the form 2q where q is an odd prime?  Can you find a pattern to ap-1º1 (mod p)?

9.     Can you find pairs of numbers a and p with p prime and (a,p)=1 such that ap-1º(mod p2)?

10. The Fibonacci sequence is defined as follows: F0=1, F1=1, and for n>1 Fn=Fn-1+Fn-2.  Which terms in the sequence are divisible by 3? By 5? By 7? Can you find a pattern or a theorem?

11. The Fibonacci sequence is defined in #10 above.  Reduce the sequence mod m for some different values of m.  What do you find?

12. Find all the Carmichael numbers less than 10,000,000.

13. Find all strong pseudoprimes to base 2 less than 100,000.

14. Consider the set .  Define multiplication as usual.  What are the prime numbers in this set?  Is 2 a prime?  Is 3 a prime?

15. Consider the Gaussian Integers, .  Multiplication is defined as for complex numbers. What are the primes in this set? Is 2 a prime?

16. Explore how many ways a number can be written as the sum of two perfect squares.

17. Suppose that nj(n)=mj(m) for two integers m and n.  Does it follow that m=n?

18. Write a computer program to implement Pollard’s Rho method for factorization.

19. Find infinitely many solutions to the following example of Pell’s equation: x2-2y2=±1.

20. Implement the continued fraction algorithm for factorization.

21. When are the numbers known as repunits prime? A repunit is a number whose expression consists only of 1’s, for example 11 or 111.

22. The Fermat numbers are numbers of the form .  For which n are these numbers primes?  Why is 2n+1 never a prime unless n is a power of 2?

23. We saw that a perfect number is an integer n such that s(n)=2n.  A multiply perfect number is one for which s(n)=kn for some integer k larger that 2.  Find some 3 perfect numbers (k=3).

24. Amicable numbers are pairs of integers m and n such that s(m)=n and s(n)=m.  Find some amicable numbers.

25. Iterate the phi function. Start with an integer n and compute j(n), j(j(n)), j(j(j(n))) etc.  What happens?

26. Are there any odd, composite numbers k such that if k is the smallest spsp(a) then k is smaller that a?

27. Are there any numbers k such that the smallest spsp(k) are much larger than k.

 

Presentation Topics

1.     Peano’s Axiomatization of Arithmetic.

2.     Surreal Numbers

3.     Sophie Germain’s advances towards a proof of Fermat’s Last Theorem.

4.     Pythagorean Triples.

5.     Pell’s Equation

6.     Continued Fractions

 

 

Textbook Problems

1.     pg. 19 #46

2.     pg. 19 #47

3.     pg. 32 #34

4.     pg. 34 #48

5.     pg. 32 #35

6.     pg. 58 #46

7.     pg. 59 #52