MIDTERM EXAM REVIEW MATH 115: NUMBER THEORY

Problem 1. Show that the set S ⊂ ℝ of positive irrational numbers is not wellordered.

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Problem 2. Find a solution x ∈ ℤ/65ℤ to

x^{2} + 1 ≡ 0 (mod 65).

How many distinct solutions x ∈ ℤ/65ℤ are there to this congruence?

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Problem 3. Show that if p, q are distinct primes, then p^{q−1} + q^{p−1} ≡ 1 (mod pq).

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Problem 4. An integer x is randomly chosen between 100 and 1000. Estimate the probability that x is prime. [Hint: log(10) ≈ 2.5.]

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Problem 5. For which integers n is it true that n − 2 divides 2n^{2} − 1?

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Problem 6. Find a solution x ∈ ℤ to the congruence

x^{2} + 1 ≡ 0 (mod 101^{2}).

[Hint: 101 is prime.]