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Number Theory 2017-2018 Example Sheet 4
Module: Number theory
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University: University of Cambridge
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Michaelmas Term 2017
Number Theory: Example Sheet 4 of 4
Throughout this sheet, Ndenotes an odd positive integer.
1. Let dand mbe positive integers such that dis not a square and such that m6√d.
Prove that if xand yare positive integers satisfying x2
−dy2=mthen x/y is a
convergent of √d.
2. Determine which of the equations x2
−31y2= 1, x2
−31y2= 4 and x2
−31y2= 5
are soluble in positive integers xand y. For each that is soluble, exhibit at least
one solution.
3. Find two solutions in positive integers xand yof the equation x2
−dy2= 1 when
d= 3, 7, 13, 19.
4. Find all bases for which 39 is an Euler pseudoprime.
5. Let Nbe an odd composite integer.
(i) Show that if Nis a Carmichael number, then Nis square-free.
(ii) Show that Nis a Carmichael number if and only if Nis square-free and p−1
divides N−1 for every prime pdividing N.
(iii) Show that if Nis a Carmichael number, then Nis the product of at least
three distinct primes.
(iv) Find the smallest Carmichael number.
6. Let N= (6t+ 1)(12t+ 1)(18t+ 1), where tis a positive integer such that 6t+ 1,
12t+ 1 and 18t+ 1 are all prime numbers. Prove that Nis a Carmichael number.
Use this construction to find three Carmichael numbers. (You will need to come
up with a better method than simply trying t= 1,2,3, . . ..)
7. Prove that there are 36 bases for which 91 is a pseudoprime. More generally, show
that if pand 2p−1 are both prime numbers, then N=p(2p−1) is a pseudoprime
for precisely half of all bases.
8. Let N= 561. Find the number of bases bfor which Nis an Euler pseudoprime.
Show that there are precisely 10 bases for which Nis a strong pseudoprime.
9. Let pbe a prime greater than 5. Prove that N= (4p+ 1)/5 is a composite integer.
Prove that Nis a strong pseudoprime to the base 2.
10. Assume that nis an integer greater than 1 such that Fn= 22n+ 1 is composite
(n= 5, . . .). Prove that Fnis a pseudoprime to the base 2.
11. Prove that if Nhas a factor which is within 4
√Nof √N, then Fermat factorisation
must work on the first try.
12. Use Fermat factorisation to factor the integers 8633, 809009, and 92296873.
a.j.scholl@dpmms.cam.ac.uk - 1 - 22 November 2017
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