Download Some Old Computational Number Theory Comp Problems

Some Old Computational Number Theory Comp Problems
1. Define a strong pseudoprime to the base b and prove that no integer n > 1 is a strong
pseudoprime to every base b with 1 ≤ b ≤ n and gcd(b, n) = 1.
2. (a) Prove that 2465 is an absolute pseudoprime.
(b) Is 2465 a strong pseudoprime to the base 2? Justify your answer. (Hint: Work modulo
the prime factors of 2465 rather than modulo 2465.)
3. (a) State the Proth-Pocklington-Lehmer primality test.
(b) Prove the Proth-Pocklington-Lehmer primality test.
4. Demonstrate what MAPLE’s version of the Lucas-Lehmer test is doing by checking the
primality of n = 37 using P = 3.
5. Show that
|{(a, b) ∈ Z2 : 1 ≤ a ≤ N, 1 ≤ b ≤ N, gcd(a, b) > log log log N }|
= 0.
N →∞
N2
√
6. Prove that most positive integers n have a prime factor > n. More specifically, show that
√
|{n ≤ x : ∃ a prime p > n such that p|n}|
1
lim inf
> .
x→∞
x
2
lim
7. Given n = 12371, describe precisely how to use Dixon’s Factoring Algorithm and the following information to find a nontrivial factor of n. You do not need to come up with a
factor of n, but use Dixon’s Factoring Algorithm to reduce coming up with a factor of n to
the computation of gcd(a, n) where you tell me rather precisely what the value of a is (it
should involve multiplication and addition of specific numbers). It is possible that the a you
choose will not produce a factorization of n; in the algorithm one might need to try more
than one value of a. You need only give me one reasonable choice for a. Use the following
information where all congruences shown are modulo n:
1162 ≡ 5 × 7 × 31,
1702 ≡ 2 × 33 × 7 × 11,
1362 ≡ 53 × 72 ,
1732 ≡ 3 × 7 × 13 × 19,
1592 ≡ 72 × 11,
1842 ≡ 2 × 3 × 72 × 31.
8. Let n = 1377989.
The quadratic sieve algorithm is used to factor n as follows. It is deter√
mined that [ n] = 1173, and s(a) = a2 mod n is computed for 1174 ≤ a ≤ 2000. (We
ignore here the possibility of s(a) being negative and only use positive residues.) We factor
each such s(a) and discard s(a) if it has a prime factor > 20. We obtain the following data:
Values for a
1192
1679
1688
1954
a2 (mod n)
53 × 73
32 × 72 × 11 × 13
2 × 33 × 7 × 13 × 19
2 × 3 × 7 × 113 × 19
Continue the quadratic sieve algorithm by finding numbers A and B for which gcd(A, B) is
reasonably a non-trivial divisor of n. You do not need to compute the gcd(A, B). Furthermore, you do not need to give A and B explicitly. Specifically, A may be left in the form
A = u × v × w × x × y mod n (the number of variables written here is not significant), and
similarly for B.
9. Let f (x) = x6 + x5 + x4 + x3 + x2 + x + 1. The purpose of this problem is to factor f (x)
modulo 2 using Berlekamp’s algorithm. Working modulo 2, we compute a certain matrix A
and then B = A − I. The result of this computation is (in the field of arithmetic modulo 2)


0 0 0 1 ∗ 0


 0 1 0 1 ∗ 0 




 0 1 1 1 ∗ 0 

B =A−I =
 0 0 0 0 ∗ 1 ,




 0 0 1 1 ∗ 0 


0 0 0 1 ∗ 1
where the elements of the fifth column have been replaced by asterisks. Using Berlekamp’s
algorithm and what has been stated here, find a polynomial g(x) of degree ≤ 4 such that
when gcd(f (x), g(x)) is computed modulo 2, the result is a non-trivial factor of f (x) modulo 2. You do not need to justify the entries in B above, but you should indicate what the fifth
column is and clarify how you obtain it. You may want to check other columns given with
your approach to see if you are computing B correctly. You should also indicate g(x) explicitly, explain how you obtain g(x) from B and then make use of g(x) to obtain a complete
factorization of f (x) modulo 2.
10. Let
n
Y
f (x) = an (x − αj )
and
w(x) = an
j=1
Y
(x − αj )
1≤j≤n
|αj |>1
Y
(αj x − 1).
1≤j≤n
|αj |≤1
Recall that f˜(x) = xn f (1/x) and w(x)
e
= xn w(1/x).
(a) Explain why w(x)w(x)
e
= f (x)f˜(x).
(b) Prove M (f ) ≤ kf k (where M (f ) is the Mahler measure of f ).
11. (a) For f (x) ∈ R[x], prove M (f ) ≤ kf k.
(b) For f (x) ∈ R[x], prove kf k ≤ 2deg f M (f ).
(c) Let f (x) and g(x) be polynomials in Z[x] such that g(x)|f (x). Prove kgk ≤ 2deg g kf k.