# Smallest primes

Most Lisp implementations include a primitive called `runtime` that returns an integer that specifies the amount of time the system has been running (measured, for example, in microseconds). The following `timed-prime-test` procedure, when called with an integer `n` , prints `n` and checks to see if `n` is prime. If it's prime, the procedure prints three asterisks followed by the amount of time used in performing the test.

``````(define (timed-prime-test n)
(newline)
(display n)
(start-prime-test n (runtime)))
(define (start-prime-test n start-time)
(if (prime? n)
(report-prime (- (runtime) start-time))))
(define (report-prime elapsed-time)
(display " *** ")
(display elapsed-time))
``````

Using this procedure, write a procedure `search-for-primes` that checks the primality of consecutive odd integers in a specified range. Use your procedure to find the three smallest primes larger than 1000; larger than 10,000; larger than 100,000; larger than 1,000,000. Note the time needed to test each prime. Since the testing algorithm has order of growth of Θ(√n), you should expect that testing for primes around 10,000 should take about √10 times as long as testing for primes around 1000. Do your timing data bear this out? How well do the data for 100,000 and 1,000,000 support the √n prediction? Is your result compatible with the notion that programs on your machine run in time proportional to the number of steps required for the computation?