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By Waclaw Sierpinski

ISBN-10: 0444000712

ISBN-13: 9780444000712

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Extra resources for 250 problems in elementary number theory

Example text

Divisibility and Primes 16. Compute the convergents pn /qn of the simple continued fraction 1, 1, 1, 1, 1, 1, 1 . Observe that pn fn+1 = qn fn for n = 0, 1, . . , 6. 17. Prove that f1 + f2 + · · · + fn = fn+2 − 1 for all positive integers n. 18. Prove that fn+1 fn−1 − fn2 = (−1)n for all positive integers n. 19. Prove that fn = fk+1 fn−k + fk fn−k−1 for all k = 0, 1, . . , n. Equivalently, fn = fn−1 + fn−2 = 2fn−2 + fn−3 = 3fn−3 + 2fn−4 = 5fn−4 + 3fn−5 · · · . 20. Prove that fn divides f n for all positive integers .

Prove that (fn , fn+1 ) = 1 for all nonnegative integers n. In Exercises 16–23, fn denotes the nth Fibonacci number. 24 1. Divisibility and Primes 16. Compute the convergents pn /qn of the simple continued fraction 1, 1, 1, 1, 1, 1, 1 . Observe that pn fn+1 = qn fn for n = 0, 1, . . , 6. 17. Prove that f1 + f2 + · · · + fn = fn+2 − 1 for all positive integers n. 18. Prove that fn+1 fn−1 − fn2 = (−1)n for all positive integers n. 19. Prove that fn = fk+1 fn−k + fk fn−k−1 for all k = 0, 1, . . , n.

Use the sieve of Eratosthenes to ﬁnd the prime numbers up to 210. Compute π(210). 2. Let N = 210. Prove that N − p is prime for every prime p such that N/2 < p < N . Find a prime number q < N/2 such that N − q is composite. 3. Let N = 105. Show that N − 2n is prime whenever 2 ≤ 2n < N . This statement is also true for N = 7, 15, 21, 45, and 75. It is not known whether N = 105 is the largest integer with this property. 4. Let N = 199. Show that N − 2n2 is prime whenever 2n2 < N . It is not known whether N = 199 is the largest integer with this property.

### 250 problems in elementary number theory by Waclaw Sierpinski

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