I was browsing through a copy of the New Turing Omnibus, and ran across the article on detecting primes.I grabbed an algorithms text, and implemented the Miller-Rabin primality test in Common Lisp.
;;; updated June 9, 2007 incorporating feedback from comments
(defun witness (a n)
(let ((b (- n 1)))
(loop for i from (integer-length b) downto 0
for d = 1 then (mod (* d d) n)
for x = d
do
(when (and (= d 1) (/= x 1) (/= x (- n 1)))
(return-from witness t))
(when (logbitp i b)
(setf d (mod (* d a) n)))
finally (return (/= d 1)))))
(defun primep (n &optional (s 50))
"Miller-Rabin primality test."
(dotimes (i s t)
(let ((w (1+ (random (- n 1)))))
(when (witness w n)
(return-from primep nil)))))
Some results:
; the 15th <a href="http://en.wikipedia.org/wiki/Mersenne_prime">Mersenne Prime</a>
CL-USER> (time (primep (1- (expt 2 1279))))
Evaluation took:
3.043 seconds of real time
3.034062 seconds of user run time
0.005775 seconds of system run time
[Run times include 0.071 seconds GC run time.]
0 calls to %EVAL 0 page faults and 151,872,008 bytes consed.
T
CL-USER> (1- (expt 2 1279))
1040793219466439908192524032736408553861526224726670480531911235
0403608059673362980122394417323241848424216139542810077913835662
4832346490813990660567732076292412950938922034577318334966158355
0472959420547689811211693677147548478866962513844382602917323488
8531116082853841658502825560466622483189091880184706822220314052
1026698435488732958028878050869736186900714720710555703168729087
; compare with list of first 1000 primes
; at http://en.wikipedia.org/wiki/List_of_prime_numbers
CL-USER> (loop for i from 1000 to 2000
if (primep i) collect i)
(1009 1013 1019 1021 1031 1033 1039 1049 1051 1061 1063 1069 1087
1091 1093 1097 1103 1109 1117 1123 1129 1151 1153 1163 1171 1181
1187 1193 1201 1213 1217 1223 1229 1231 1237 1249 1259 1277 1279
1283 1289 1291 1297 1301 1303 1307 1319 1321 1327 1361 1367 1373
1381 1399 1409 1423 1427 1429 1433 1439 1447 1451 1453 1459 1471
1481 1483 1487 1489 1493 1499 1511 1523 1531 1543 1549 1553 1559
1567 1571 1579 1583 1597 1601 1607 1609 1613 1619 1621 1627 1637
1657 1663 1667 1669 1693 1697 1699 1709 1721 1723 1733 1741 1747
1753 1759 1777 1783 1787 1789 1801 1811 1823 1831 1847 1861 1867
1871 1873 1877 1879 1889 1901 1907 1913 1931 1933 1949 1951 1973
1979 1987 1993 1997 1999)
Lisp’s built-in bignums makes playing with this sort of thing easy and fun.