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prime numbers

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    This subchapter looks at prime numbers.

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stub section

    This subchapter is a stub section. It will be filled in with instructional material later. For now it serves the purpose of a place holder for the order of instruction.

    Professors are invited to give feedback on both the proposed contents and the propsed order of this text book. Send commentary to Milo, PO Box 1361, Tustin, California, 92781, USA.

prime numbers

defined

    Prime numbers are integers greater than one that are only evenly divisible (no remainder) by one and themselves. That is, their only integer factors are one (1) and the number itself. rime numbers have no integer divsors other than one and themselves. yes, I have hinted at several alternative definitions. (NOTE: Some mathematicians include one as a prime number.)

    For example, some of the small prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, and 199.

    Four is not a prime because it is divisible by two. Six is divisible by both two and three. Any even number is divisible by two. Nine is divisible by three.

    An integer greater than one is prime if its only positive divisors are itself and one.

    The reason that negative primes are excluded is because prime numbers were discovered before negative numbers and a whole lot of very interesting (and sometimes useful) patterns and properties involving prime numbers had already been discovered. Many of these patterns were not true for negative numbers. To maintain the usefulness of prime numbers, they were restricted to only pisitive numbers. Zero was discovered even later and the same principle applied for its exclusion. One is excluded because some of the useful properties don’t apply to one.

primality test

    A primality test is an algorithm for determining if a number (or group of numbers) is prime or composite.

trial division

    A suboptimal, but common, method for finding prime numbers is trial division. We simply try to divide the number being tested by every integer less than that number and see if any of the divisions produce integer results (which indicates a composite number). We might consider this method (or a variation) if our goal is to identify every factor of a particular composite number.

a bad example algoithm

    While looking for something completely unrelated I came across a homework answer website that gave a COBOL program that would determine if a number was a prime number or not. I won’t name the site because there is no need to embarass one particular person. I won’t go into a deep analysis of the coding mistakes because any of those could easily be fixed, even if only by trial and error.

    Of concern to us as students of software engineering is the choice of algorithm and how the algorithm might be made more efficient.

    The following is a pseudo-code version of the COBOL program. I’d reprint the original, but COBOL is very wordy and space consuming and why waste all of that space on a bad example. Further, not a whole lot of people know COBOL.

    INPUT a number (input_number);

    failure_flag := FALSE;
    DO next_possible_divisor FROM 2 UNTIL next_possible_divisor >= input_number BY STEPS of 1;
      BEGIN
        test_result := input_number MOD next_possible_divisor;
        IF (test_result == 0)
          THEN failure_flag := TRUE;
      END;
    IF (failure_flag == TRUE)
      THEN
        PRINT(input_number, ' is not a prime');
      ELSE
        PRINT(input_number, ' is a prime');
    END;

    Note that while this version initializes the flag, the original did not actually initialize its flag!

    The program uses the MOD or modular function or operation to test for whether or not a next_possible_divisor is a integral factor of the input_number. If the remainder is zero, then it is a factor and we don’t have a prime.

stopping

    If we reach the input_number, then we can stop. We didn’t need to test for greater than or equal to because we will stop at equality.

    Another obvious improvement would be to stop as soon as we have failure. Once we have determined the number isn’t a prime, we really don’t need to make any more tests. This can remove a whole bunch of tests, especially if we fail on a number such as two, three, or five.

remove even numbers

    We can also immediately remove all of the even numbers other than two from our test. This will immediately cut the number of tests almost in half.

    Becuase we know that two is the only even number that is also a prime number, we can also skip all tests of dividing (or modulo) for any number that is even. After identifying two, we never need to do any more tests with any even numbers, either as a candidate for a possible prime or as a possible divide test for a prime.

square roots

    Now, if we think about what numbers can evenly divide another number, we can cut out another large batch of tests. Numbers that aren’t prime are called composite numbers. Let’s look at a few composite numbers.

    Four is divisible by 1, 2, and 4 (1·4, 2·2, and 4·1).

    Six is divisible by 1, 2, 3, and 6 (1·6, 2·3, 3·2, and 6·1).

    Eight is divisible by 1, 2, 4, and 8 (1·8, 2·4, 4·2, and 8·1).

    Nine is divisible by 1, 3, and 9 (1·9, 3·3, 9·1).

    If you happen to notice the bold faced combinations, you might notice a special pattern about divisors.

    There are never any integer divisors that are greater than the square root of a number.

    This gives us two new tests.

    We can find the square root of our input number.

    If the square root is an integer, then we can immediately stop our test because we have already shown it isn’t a prime number.

integer floor of square root

    If the square root is a mixed number (that is, it has a fractional part), then we can use the closest integer (rounding down) as the limit for our loop. This is the integer floor function of the square root.

    Once we realize that we only need the integer floor of the square root, we can eliminate a whole bunch of square root operations. We only need to compute the square root when we skip to the next integer in the floor function sequence. And we can completely eliminate the square root computations by taking a series of squares.

    We start with 1 squared (which is one). The next in the series is two squared (which is four). We already know that both tow and three are prime numbers, but if we were to actually have a computer do the test, it can stop at one (the integer floor of both the square root of two and the square root of three).

    Our sequence of square roots that equal the integer floor of a suqare root is:

integer 1 2 3 4 5 6 7 8 9 10
square 1 4 9 16 25 36 49 64 81 100

    So, when testing the numbers from four to eight, we use the integer floor of the square root, which is two as our stopping point.

    When testing the numbers from nine to 15, we use the integer floor of the square root, which is three as our stopping point.

    When testing the numbers from 16 to 24, we use the integer floor of the square root, which is four as our stopping point.

    This pattern continues. This pattern is called perfect squares.

    Stopping at the integer floor of the square root is a much more useful optimization than removing all even numbers other than two (which saved about half our time).

    Consider the case of 101 (which is prime). The square root is a little more than 10 (the square root of 100 is 10). We can stop testing at 10. 11 times 9 is simply the reverse of the test 9 times 11, which we would have already done. This elminates about 90% of the possible tests. The efficiency (as a percentage) goes up as we test larger and larger numbers.

    Consider the case of eight. Once we determineed that eight was divisible by two, we didn’t need to check for any other multiple of two. We didn’t actually have to check to see if it was divisible by four. This was the basis of our earlier optimization of throwing out all even numbers other than two.

generating the perfect squares

    We can generate the sequence of integer floors of square roots by taking each integer in turn and multiplying by itself.

    Another optimizationis to use simple additions.

    The sequence of perfect squares is generated by adding the consecutive odd integers.

    The first odd integer is one and its perfect square is one.

    The first two odd integers are one and three, which have a sum of four, the perfect square of two.

    The first three odd integers are one, three, and five, which have a sum of nine, the perfect square of three.

    The first four odd integers are one, three, five, and seven, which have a sum of 16, which is the perfect square of four.

    This pattern continues forever. The sum of the first n positive integers is n2.

    If we keep track of where we are in the sequence, we can simply determine the next odd integer and add it to our running total and use that information for determining our stopping point.

skip composite numbers

    What other optimizations can we do?

    If we happen to be looking for a list of prime numbers rather than simply testing one arbitrary number, we can save a list of all the prime numbers we’ve already found.

    Why did we skip all even numbers other than two? Because if a number is divisible by any multiple of two (4, 6, 8, 10, …), then it is also divisible by two. No need to perform those tests.

    This principle applies to any other prime number.

    Consider 81. We don’t need to test to see that 81 is divisible by nine (nine times nine equals 81) because three is a factor of nine and by the time we reached nine we would have already determined that 81 was also divisible by three.

    We can leave all composite numbers out of our test for prime numbers. Of course, this means we need to know whether or not the next possible divisor is a prime or composite number. Hence, the optimization of keeping a list of prime numbers. As we determine each prime number, we can run our test for the next larger prime number using only the prime numbers we’ve already discovered.

6k ± 1

    All primes (other than 2 and 3) are of the form 6k ± 1.

    We can use this fact to increase the speed of our check. First we check to see if the number is divisble by either two (2) or three (3), then only check additional numbers less than the square root of n that are in the form of 6k ± 1. This method is about three times as fast as checking every integer less than the square root of n.

    We can also use a check based on coprimes to further speed up the trial division method.

    Leonard Adleman and Huang presented an errorless (but expected polynomial-time) variant of the elliptic curve primality test. Unlike the other probabilistic tests, this algorithm produces a primality certificate, and thus can be used to prove that a number is prime. The algorithm is prohibitively slow in practice.

fast deterministic tests

    Near the beginning of the 20th century, it was shown that a result of Fermat’s little theorem could be used to test for primality. This resulted in the Pocklington primality test. However, as this test requires a partial factorization of n - 1 the running time was still quite slow in the worst case. The first deterministic primality test significantly faster than the naive methods was the cyclotomy test; its runtime can be proven to be O((log n)c log log log n), where n is the number to test for primality and c is a constant independent of n. Many further improvements were made, but none could be proven to have polynomial running time. (Note that running time is measured in terms of the size of the input, which in this case is ~ log n, that being the number of bits needed to represent the number n.) The elliptic curve primality test can be proven to run in O((log n)6), but only if some still unproven (but widely assumed to be true) statements of analytic number theory are used. Similarly, under the generalized Riemann hypothesis, the Miller-Rabin test can be turned into a deterministic version (called Miller’s test) with runtime O((log n)4). In practice, this algorithm is slower than the other two for sizes of numbers that can be dealt with at all. Because the implementation of these methods is rather difficult and creates a risk of programming errors, the slower but simpler tests are often preferred. In 2002 the first provably polynomial time test for primality was invented by Manindra Agrawal, Neeraj Kayal and Nitin Saxena. The AKS primality test, runs in O((log n)12) (improved to O((log n)7.5) in the published revision of their paper), which can be further reduced to O((log n)6) if the Sophie Germain conjecture is true. Subsequently, Lenstra and Pomerance presented a version of the test which runs in time O((log n)6) unconditionally.

Sieve of Eratosthenes

    The major optimization is to change the algorithm to a more efficient method.

    The famous prime number finding algorithm is the Sieve of Eratosthenes, named for Eratosthenes of Cyrene, a Greek mathematician. In the book Introduction to Arithmetic Nicomachus attributed this algorithm to Eratosthenes.

    The original algorithm was to create a list of integers (up to a limit), then repeatedly mark off composite numbers. For example, start with 2 (the first prime), and then mark off every other number (all of the even numbers after 2). Then start with 3 (the second prime) and mark off every third number (the multiples of three). Use the next prime number (5) and mark off every fifth number (the multiples of five). Continue with every prime until you have marked off all of the composite numbers below your limit and left all of the prime numbers below your limit unmarked.

incremental sieve

    We can improve on the original Sieve of Eratosthenes by eliminating the need for a limit (or upper bound). We interleave the marking of composites with the discovery of primes and continue up the list doing all of the composite marking at once.

better algorithms

    Are there more optimizations? Yes!

    The important principle (once again) is that we can dramatically improve the efficiency of our programs by looking at the fundamental algorithm we are using.

    Many beginning programmers point out that some of the steps that make a program clear and easy to read have a small built-in inefficiency. Cleaning up those small inefficiencies makes only a very small improvement in the total efficiency of the program (except for a few bottleneck cases). This small improvement in efficiency is not worth the headaches that occur when someone else has to come along and change your program.

    With the increasing speed and power of processors (Moore’s Law) the time saved from little tricks is trivial to the point of being insignificant.

    The real improvements in efficiency come from finding better methods (better algorithms).

    Please keep your programs readable, because the small amount of processing time you save is trivial compared to the large amount of extra time when the next programmer has to make an inevitable change. Successful programs have lasted for decades. That’s why there are still early COBOL and FORTRAN programs in use.

    For a very interesting web site about prime numbers see primes.utm.edu.

first 10,000 prime numbers

(the 10,000th prime is 104,729)

      2      3      5      7     11     13     17     19     23     29
     31     37     41     43     47     53     59     61     67     71
     73     79     83     89     97    101    103    107    109    113
    127    131    137    139    149    151    157    163    167    173
    179    181    191    193    197    199    211    223    227    229
    233    239    241    251    257    263    269    271    277    281
    283    293    307    311    313    317    331    337    347    349
    353    359    367    373    379    383    389    397    401    409
    419    421    431    433    439    443    449    457    461    463
    467    479    487    491    499    503    509    521    523    541
    547    557    563    569    571    577    587    593    599    601
    607    613    617    619    631    641    643    647    653    659
    661    673    677    683    691    701    709    719    727    733
    739    743    751    757    761    769    773    787    797    809
    811    821    823    827    829    839    853    857    859    863
    877    881    883    887    907    911    919    929    937    941
    947    953    967    971    977    983    991    997   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   2003   2011   2017   2027   2029   2039   2053
   2063   2069   2081   2083   2087   2089   2099   2111   2113   2129
   2131   2137   2141   2143   2153   2161   2179   2203   2207   2213
   2221   2237   2239   2243   2251   2267   2269   2273   2281   2287
   2293   2297   2309   2311   2333   2339   2341   2347   2351   2357
   2371   2377   2381   2383   2389   2393   2399   2411   2417   2423
   2437   2441   2447   2459   2467   2473   2477   2503   2521   2531
   2539   2543   2549   2551   2557   2579   2591   2593   2609   2617
   2621   2633   2647   2657   2659   2663   2671   2677   2683   2687
   2689   2693   2699   2707   2711   2713   2719   2729   2731   2741
   2749   2753   2767   2777   2789   2791   2797   2801   2803   2819
   2833   2837   2843   2851   2857   2861   2879   2887   2897   2903
   2909   2917   2927   2939   2953   2957   2963   2969   2971   2999
   3001   3011   3019   3023   3037   3041   3049   3061   3067   3079
   3083   3089   3109   3119   3121   3137   3163   3167   3169   3181
   3187   3191   3203   3209   3217   3221   3229   3251   3253   3257
   3259   3271   3299   3301   3307   3313   3319   3323   3329   3331
   3343   3347   3359   3361   3371   3373   3389   3391   3407   3413
   3433   3449   3457   3461   3463   3467   3469   3491   3499   3511
   3517   3527   3529   3533   3539   3541   3547   3557   3559   3571
   3581   3583   3593   3607   3613   3617   3623   3631   3637   3643
   3659   3671   3673   3677   3691   3697   3701   3709   3719   3727
   3733   3739   3761   3767   3769   3779   3793   3797   3803   3821
   3823   3833   3847   3851   3853   3863   3877   3881   3889   3907
   3911   3917   3919   3923   3929   3931   3943   3947   3967   3989
   4001   4003   4007   4013   4019   4021   4027   4049   4051   4057
   4073   4079   4091   4093   4099   4111   4127   4129   4133   4139
   4153   4157   4159   4177   4201   4211   4217   4219   4229   4231
   4241   4243   4253   4259   4261   4271   4273   4283   4289   4297
   4327   4337   4339   4349   4357   4363   4373   4391   4397   4409
   4421   4423   4441   4447   4451   4457   4463   4481   4483   4493
   4507   4513   4517   4519   4523   4547   4549   4561   4567   4583
   4591   4597   4603   4621   4637   4639   4643   4649   4651   4657
   4663   4673   4679   4691   4703   4721   4723   4729   4733   4751
   4759   4783   4787   4789   4793   4799   4801   4813   4817   4831
   4861   4871   4877   4889   4903   4909   4919   4931   4933   4937
   4943   4951   4957   4967   4969   4973   4987   4993   4999   5003
   5009   5011   5021   5023   5039   5051   5059   5077   5081   5087
   5099   5101   5107   5113   5119   5147   5153   5167   5171   5179
   5189   5197   5209   5227   5231   5233   5237   5261   5273   5279
   5281   5297   5303   5309   5323   5333   5347   5351   5381   5387
   5393   5399   5407   5413   5417   5419   5431   5437   5441   5443
   5449   5471   5477   5479   5483   5501   5503   5507   5519   5521
   5527   5531   5557   5563   5569   5573   5581   5591   5623   5639
   5641   5647   5651   5653   5657   5659   5669   5683   5689   5693
   5701   5711   5717   5737   5741   5743   5749   5779   5783   5791
   5801   5807   5813   5821   5827   5839   5843   5849   5851   5857
   5861   5867   5869   5879   5881   5897   5903   5923   5927   5939
   5953   5981   5987   6007   6011   6029   6037   6043   6047   6053
   6067   6073   6079   6089   6091   6101   6113   6121   6131   6133
   6143   6151   6163   6173   6197   6199   6203   6211   6217   6221
   6229   6247   6257   6263   6269   6271   6277   6287   6299   6301
   6311   6317   6323   6329   6337   6343   6353   6359   6361   6367
   6373   6379   6389   6397   6421   6427   6449   6451   6469   6473
   6481   6491   6521   6529   6547   6551   6553   6563   6569   6571
   6577   6581   6599   6607   6619   6637   6653   6659   6661   6673
   6679   6689   6691   6701   6703   6709   6719   6733   6737   6761
   6763   6779   6781   6791   6793   6803   6823   6827   6829   6833
   6841   6857   6863   6869   6871   6883   6899   6907   6911   6917
   6947   6949   6959   6961   6967   6971   6977   6983   6991   6997
   7001   7013   7019   7027   7039   7043   7057   7069   7079   7103
   7109   7121   7127   7129   7151   7159   7177   7187   7193   7207
   7211   7213   7219   7229   7237   7243   7247   7253   7283   7297
   7307   7309   7321   7331   7333   7349   7351   7369   7393   7411
   7417   7433   7451   7457   7459   7477   7481   7487   7489   7499
   7507   7517   7523   7529   7537   7541   7547   7549   7559   7561
   7573   7577   7583   7589   7591   7603   7607   7621   7639   7643
   7649   7669   7673   7681   7687   7691   7699   7703   7717   7723
   7727   7741   7753   7757   7759   7789   7793   7817   7823   7829
   7841   7853   7867   7873   7877   7879   7883   7901   7907   7919
   7927   7933   7937   7949   7951   7963   7993   8009   8011   8017
   8039   8053   8059   8069   8081   8087   8089   8093   8101   8111
   8117   8123   8147   8161   8167   8171   8179   8191   8209   8219
   8221   8231   8233   8237   8243   8263   8269   8273   8287   8291
   8293   8297   8311   8317   8329   8353   8363   8369   8377   8387
   8389   8419   8423   8429   8431   8443   8447   8461   8467   8501
   8513   8521   8527   8537   8539   8543   8563   8573   8581   8597
   8599   8609   8623   8627   8629   8641   8647   8663   8669   8677
   8681   8689   8693   8699   8707   8713   8719   8731   8737   8741
   8747   8753   8761   8779   8783   8803   8807   8819   8821   8831
   8837   8839   8849   8861   8863   8867   8887   8893   8923   8929
   8933   8941   8951   8963   8969   8971   8999   9001   9007   9011
   9013   9029   9041   9043   9049   9059   9067   9091   9103   9109
   9127   9133   9137   9151   9157   9161   9173   9181   9187   9199
   9203   9209   9221   9227   9239   9241   9257   9277   9281   9283
   9293   9311   9319   9323   9337   9341   9343   9349   9371   9377
   9391   9397   9403   9413   9419   9421   9431   9433   9437   9439
   9461   9463   9467   9473   9479   9491   9497   9511   9521   9533
   9539   9547   9551   9587   9601   9613   9619   9623   9629   9631
   9643   9649   9661   9677   9679   9689   9697   9719   9721   9733
   9739   9743   9749   9767   9769   9781   9787   9791   9803   9811
   9817   9829   9833   9839   9851   9857   9859   9871   9883   9887
   9901   9907   9923   9929   9931   9941   9949   9967   9973  10007
  10009  10037  10039  10061  10067  10069  10079  10091  10093  10099
  10103  10111  10133  10139  10141  10151  10159  10163  10169  10177
  10181  10193  10211  10223  10243  10247  10253  10259  10267  10271
  10273  10289  10301  10303  10313  10321  10331  10333  10337  10343
  10357  10369  10391  10399  10427  10429  10433  10453  10457  10459
  10463  10477  10487  10499  10501  10513  10529  10531  10559  10567
  10589  10597  10601  10607  10613  10627  10631  10639  10651  10657
  10663  10667  10687  10691  10709  10711  10723  10729  10733  10739
  10753  10771  10781  10789  10799  10831  10837  10847  10853  10859
  10861  10867  10883  10889  10891  10903  10909  10937  10939  10949
  10957  10973  10979  10987  10993  11003  11027  11047  11057  11059
  11069  11071  11083  11087  11093  11113  11117  11119  11131  11149
  11159  11161  11171  11173  11177  11197  11213  11239  11243  11251
  11257  11261  11273  11279  11287  11299  11311  11317  11321  11329
  11351  11353  11369  11383  11393  11399  11411  11423  11437  11443
  11447  11467  11471  11483  11489  11491  11497  11503  11519  11527
  11549  11551  11579  11587  11593  11597  11617  11621  11633  11657
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free music player coding example

    Coding example: I am making heavily documented and explained open source code for a method to play music for free — almost any song, no subscription fees, no download costs, no advertisements, all completely legal. This is done by building a front-end to YouTube (which checks the copyright permissions for you).

    View music player in action: www.musicinpublic.com/.

    Create your own copy from the original source code/ (presented for learning programming).


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Because I no longer have the computer and software to make PDFs, the book is available as an HTML file, which you can convert into a PDF.

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free computer programming text book project

Building a free downloadable text book on computer programming for university, college, community college, and high school classes in computer programming.

If you like the idea of this project,
then please donate some money.

send donations to:
Milo
PO Box 1361
Tustin, California 92781

Supporting the entire project:

    If you have a business or organization that can support the entire cost of this project, please contact Pr Ntr Kmt (my church)

more information on donating

Some or all of the material on this web page appears in the
free downloadable college text book on computer programming.


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Made with Macintosh

    This web site handcrafted on Macintosh computers using Tom Bender’s Tex-Edit Plus and served using FreeBSD .

Viewable With Any Browser


    †UNIX used as a generic term unless specifically used as a trademark (such as in the phrase “UNIX certified”). UNIX is a registered trademark in the United States and other countries, licensed exclusively through X/Open Company Ltd.

    Names and logos of various OSs are trademarks of their respective owners.

    Copyright © 2010, 2011, 2012, 2013 Milo

    Created: December 11, 2010

    Last Updated: April 24, 2013


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