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prime numbers
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This subchapter looks at prime numbers.
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This subchapter looks at prime numbers.
free computer programming text book projecttable of contents
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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 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.
A primality test is an algorithm for determining if a number (or group of numbers) is prime or composite.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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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
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877 881 883 887 907 911 919 929 937 941
947 953 967 971 977 983 991 997 1009 1013
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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
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9461 9463 9467 9473 9479 9491 9497 9511 9521 9533
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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
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10753 10771 10781 10789 10799 10831 10837 10847 10853 10859
10861 10867 10883 10889 10891 10903 10909 10937 10939 10949
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14951 14957 14969 14983 15013 15017 15031 15053 15061 15073
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15787 15791 15797 15803 15809 15817 15823 15859 15877 15881
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16487 16493 16519 16529 16547 16553 16561 16567 16573 16603
16607 16619 16631 16633 16649 16651 16657 16661 16673 16691
16693 16699 16703 16729 16741 16747 16759 16763 16787 16811
16823 16829 16831 16843 16871 16879 16883 16889 16901 16903
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17099 17107 17117 17123 17137 17159 17167 17183 17189 17191
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17789 17791 17807 17827 17837 17839 17851 17863 17881 17891
17903 17909 17911 17921 17923 17929 17939 17957 17959 17971
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18521 18523 18539 18541 18553 18583 18587 18593 18617 18637
18661 18671 18679 18691 18701 18713 18719 18731 18743 18749
18757 18773 18787 18793 18797 18803 18839 18859 18869 18899
18911 18913 18917 18919 18947 18959 18973 18979 19001 19009
19013 19031 19037 19051 19069 19073 19079 19081 19087 19121
19139 19141 19157 19163 19181 19183 19207 19211 19213 19219
19231 19237 19249 19259 19267 19273 19289 19301 19309 19319
19333 19373 19379 19381 19387 19391 19403 19417 19421 19423
19427 19429 19433 19441 19447 19457 19463 19469 19471 19477
19483 19489 19501 19507 19531 19541 19543 19553 19559 19571
19577 19583 19597 19603 19609 19661 19681 19687 19697 19699
19709 19717 19727 19739 19751 19753 19759 19763 19777 19793
19801 19813 19819 19841 19843 19853 19861 19867 19889 19891
19913 19919 19927 19937 19949 19961 19963 19973 19979 19991
19993 19997 20011 20021 20023 20029 20047 20051 20063 20071
20089 20101 20107 20113 20117 20123 20129 20143 20147 20149
20161 20173 20177 20183 20201 20219 20231 20233 20249 20261
20269 20287 20297 20323 20327 20333 20341 20347 20353 20357
20359 20369 20389 20393 20399 20407 20411 20431 20441 20443
20477 20479 20483 20507 20509 20521 20533 20543 20549 20551
20563 20593 20599 20611 20627 20639 20641 20663 20681 20693
20707 20717 20719 20731 20743 20747 20749 20753 20759 20771
20773 20789 20807 20809 20849 20857 20873 20879 20887 20897
20899 20903 20921 20929 20939 20947 20959 20963 20981 20983
21001 21011 21013 21017 21019 21023 21031 21059 21061 21067
21089 21101 21107 21121 21139 21143 21149 21157 21163 21169
21179 21187 21191 21193 21211 21221 21227 21247 21269 21277
21283 21313 21317 21319 21323 21341 21347 21377 21379 21383
21391 21397 21401 21407 21419 21433 21467 21481 21487 21491
21493 21499 21503 21517 21521 21523 21529 21557 21559 21563
21569 21577 21587 21589 21599 21601 21611 21613 21617 21647
21649 21661 21673 21683 21701 21713 21727 21737 21739 21751
21757 21767 21773 21787 21799 21803 21817 21821 21839 21841
21851 21859 21863 21871 21881 21893 21911 21929 21937 21943
21961 21977 21991 21997 22003 22013 22027 22031 22037 22039
22051 22063 22067 22073 22079 22091 22093 22109 22111 22123
22129 22133 22147 22153 22157 22159 22171 22189 22193 22229
22247 22259 22271 22273 22277 22279 22283 22291 22303 22307
22343 22349 22367 22369 22381 22391 22397 22409 22433 22441
22447 22453 22469 22481 22483 22501 22511 22531 22541 22543
22549 22567 22571 22573 22613 22619 22621 22637 22639 22643
22651 22669 22679 22691 22697 22699 22709 22717 22721 22727
22739 22741 22751 22769 22777 22783 22787 22807 22811 22817
22853 22859 22861 22871 22877 22901 22907 22921 22937 22943
22961 22963 22973 22993 23003 23011 23017 23021 23027 23029
23039 23041 23053 23057 23059 23063 23071 23081 23087 23099
23117 23131 23143 23159 23167 23173 23189 23197 23201 23203
23209 23227 23251 23269 23279 23291 23293 23297 23311 23321
23327 23333 23339 23357 23369 23371 23399 23417 23431 23447
23459 23473 23497 23509 23531 23537 23539 23549 23557 23561
23563 23567 23581 23593 23599 23603 23609 23623 23627 23629
23633 23663 23669 23671 23677 23687 23689 23719 23741 23743
23747 23753 23761 23767 23773 23789 23801 23813 23819 23827
23831 23833 23857 23869 23873 23879 23887 23893 23899 23909
23911 23917 23929 23957 23971 23977 23981 23993 24001 24007
24019 24023 24029 24043 24049 24061 24071 24077 24083 24091
24097 24103 24107 24109 24113 24121 24133 24137 24151 24169
24179 24181 24197 24203 24223 24229 24239 24247 24251 24281
24317 24329 24337 24359 24371 24373 24379 24391 24407 24413
24419 24421 24439 24443 24469 24473 24481 24499 24509 24517
24527 24533 24547 24551 24571 24593 24611 24623 24631 24659
24671 24677 24683 24691 24697 24709 24733 24749 24763 24767
24781 24793 24799 24809 24821 24841 24847 24851 24859 24877
24889 24907 24917 24919 24923 24943 24953 24967 24971 24977
24979 24989 25013 25031 25033 25037 25057 25073 25087 25097
25111 25117 25121 25127 25147 25153 25163 25169 25171 25183
25189 25219 25229 25237 25243 25247 25253 25261 25301 25303
25307 25309 25321 25339 25343 25349 25357 25367 25373 25391
25409 25411 25423 25439 25447 25453 25457 25463 25469 25471
25523 25537 25541 25561 25577 25579 25583 25589 25601 25603
25609 25621 25633 25639 25643 25657 25667 25673 25679 25693
25703 25717 25733 25741 25747 25759 25763 25771 25793 25799
25801 25819 25841 25847 25849 25867 25873 25889 25903 25913
25919 25931 25933 25939 25943 25951 25969 25981 25997 25999
26003 26017 26021 26029 26041 26053 26083 26099 26107 26111
26113 26119 26141 26153 26161 26171 26177 26183 26189 26203
26209 26227 26237 26249 26251 26261 26263 26267 26293 26297
26309 26317 26321 26339 26347 26357 26371 26387 26393 26399
26407 26417 26423 26431 26437 26449 26459 26479 26489 26497
26501 26513 26539 26557 26561 26573 26591 26597 26627 26633
26641 26647 26669 26681 26683 26687 26693 26699 26701 26711
26713 26717 26723 26729 26731 26737 26759 26777 26783 26801
26813 26821 26833 26839 26849 26861 26863 26879 26881 26891
26893 26903 26921 26927 26947 26951 26953 26959 26981 26987
26993 27011 27017 27031 27043 27059 27061 27067 27073 27077
27091 27103 27107 27109 27127 27143 27179 27191 27197 27211
27239 27241 27253 27259 27271 27277 27281 27283 27299 27329
27337 27361 27367 27397 27407 27409 27427 27431 27437 27449
27457 27479 27481 27487 27509 27527 27529 27539 27541 27551
27581 27583 27611 27617 27631 27647 27653 27673 27689 27691
27697 27701 27733 27737 27739 27743 27749 27751 27763 27767
27773 27779 27791 27793 27799 27803 27809 27817 27823 27827
27847 27851 27883 27893 27901 27917 27919 27941 27943 27947
27953 27961 27967 27983 27997 28001 28019 28027 28031 28051
28057 28069 28081 28087 28097 28099 28109 28111 28123 28151
28163 28181 28183 28201 28211 28219 28229 28277 28279 28283
28289 28297 28307 28309 28319 28349 28351 28387 28393 28403
28409 28411 28429 28433 28439 28447 28463 28477 28493 28499
28513 28517 28537 28541 28547 28549 28559 28571 28573 28579
28591 28597 28603 28607 28619 28621 28627 28631 28643 28649
28657 28661 28663 28669 28687 28697 28703 28711 28723 28729
28751 28753 28759 28771 28789 28793 28807 28813 28817 28837
28843 28859 28867 28871 28879 28901 28909 28921 28927 28933
28949 28961 28979 29009 29017 29021 29023 29027 29033 29059
29063 29077 29101 29123 29129 29131 29137 29147 29153 29167
29173 29179 29191 29201 29207 29209 29221 29231 29243 29251
29269 29287 29297 29303 29311 29327 29333 29339 29347 29363
29383 29387 29389 29399 29401 29411 29423 29429 29437 29443
29453 29473 29483 29501 29527 29531 29537 29567 29569 29573
29581 29587 29599 29611 29629 29633 29641 29663 29669 29671
29683 29717 29723 29741 29753 29759 29761 29789 29803 29819
29833 29837 29851 29863 29867 29873 29879 29881 29917 29921
29927 29947 29959 29983 29989 30011 30013 30029 30047 30059
30071 30089 30091 30097 30103 30109 30113 30119 30133 30137
30139 30161 30169 30181 30187 30197 30203 30211 30223 30241
30253 30259 30269 30271 30293 30307 30313 30319 30323 30341
30347 30367 30389 30391 30403 30427 30431 30449 30467 30469
30491 30493 30497 30509 30517 30529 30539 30553 30557 30559
30577 30593 30631 30637 30643 30649 30661 30671 30677 30689
30697 30703 30707 30713 30727 30757 30763 30773 30781 30803
30809 30817 30829 30839 30841 30851 30853 30859 30869 30871
30881 30893 30911 30931 30937 30941 30949 30971 30977 30983
31013 31019 31033 31039 31051 31063 31069 31079 31081 31091
31121 31123 31139 31147 31151 31153 31159 31177 31181 31183
31189 31193 31219 31223 31231 31237 31247 31249 31253 31259
31267 31271 31277 31307 31319 31321 31327 31333 31337 31357
31379 31387 31391 31393 31397 31469 31477 31481 31489 31511
31513 31517 31531 31541 31543 31547 31567 31573 31583 31601
31607 31627 31643 31649 31657 31663 31667 31687 31699 31721
31723 31727 31729 31741 31751 31769 31771 31793 31799 31817
31847 31849 31859 31873 31883 31891 31907 31957 31963 31973
31981 31991 32003 32009 32027 32029 32051 32057 32059 32063
32069 32077 32083 32089 32099 32117 32119 32141 32143 32159
32173 32183 32189 32191 32203 32213 32233 32237 32251 32257
32261 32297 32299 32303 32309 32321 32323 32327 32341 32353
32359 32363 32369 32371 32377 32381 32401 32411 32413 32423
32429 32441 32443 32467 32479 32491 32497 32503 32507 32531
32533 32537 32561 32563 32569 32573 32579 32587 32603 32609
32611 32621 32633 32647 32653 32687 32693 32707 32713 32717
32719 32749 32771 32779 32783 32789 32797 32801 32803 32831
32833 32839 32843 32869 32887 32909 32911 32917 32933 32939
32941 32957 32969 32971 32983 32987 32993 32999 33013 33023
33029 33037 33049 33053 33071 33073 33083 33091 33107 33113
33119 33149 33151 33161 33179 33181 33191 33199 33203 33211
33223 33247 33287 33289 33301 33311 33317 33329 33331 33343
33347 33349 33353 33359 33377 33391 33403 33409 33413 33427
33457 33461 33469 33479 33487 33493 33503 33521 33529 33533
33547 33563 33569 33577 33581 33587 33589 33599 33601 33613
33617 33619 33623 33629 33637 33641 33647 33679 33703 33713
33721 33739 33749 33751 33757 33767 33769 33773 33791 33797
33809 33811 33827 33829 33851 33857 33863 33871 33889 33893
33911 33923 33931 33937 33941 33961 33967 33997 34019 34031
34033 34039 34057 34061 34123 34127 34129 34141 34147 34157
34159 34171 34183 34211 34213 34217 34231 34253 34259 34261
34267 34273 34283 34297 34301 34303 34313 34319 34327 34337
34351 34361 34367 34369 34381 34403 34421 34429 34439 34457
34469 34471 34483 34487 34499 34501 34511 34513 34519 34537
34543 34549 34583 34589 34591 34603 34607 34613 34631 34649
34651 34667 34673 34679 34687 34693 34703 34721 34729 34739
34747 34757 34759 34763 34781 34807 34819 34841 34843 34847
34849 34871 34877 34883 34897 34913 34919 34939 34949 34961
34963 34981 35023 35027 35051 35053 35059 35069 35081 35083
35089 35099 35107 35111 35117 35129 35141 35149 35153 35159
35171 35201 35221 35227 35251 35257 35267 35279 35281 35291
35311 35317 35323 35327 35339 35353 35363 35381 35393 35401
35407 35419 35423 35437 35447 35449 35461 35491 35507 35509
35521 35527 35531 35533 35537 35543 35569 35573 35591 35593
35597 35603 35617 35671 35677 35729 35731 35747 35753 35759
35771 35797 35801 35803 35809 35831 35837 35839 35851 35863
35869 35879 35897 35899 35911 35923 35933 35951 35963 35969
35977 35983 35993 35999 36007 36011 36013 36017 36037 36061
36067 36073 36083 36097 36107 36109 36131 36137 36151 36161
36187 36191 36209 36217 36229 36241 36251 36263 36269 36277
36293 36299 36307 36313 36319 36341 36343 36353 36373 36383
36389 36433 36451 36457 36467 36469 36473 36479 36493 36497
36523 36527 36529 36541 36551 36559 36563 36571 36583 36587
36599 36607 36629 36637 36643 36653 36671 36677 36683 36691
36697 36709 36713 36721 36739 36749 36761 36767 36779 36781
36787 36791 36793 36809 36821 36833 36847 36857 36871 36877
36887 36899 36901 36913 36919 36923 36929 36931 36943 36947
36973 36979 36997 37003 37013 37019 37021 37039 37049 37057
37061 37087 37097 37117 37123 37139 37159 37171 37181 37189
37199 37201 37217 37223 37243 37253 37273 37277 37307 37309
37313 37321 37337 37339 37357 37361 37363 37369 37379 37397
37409 37423 37441 37447 37463 37483 37489 37493 37501 37507
37511 37517 37529 37537 37547 37549 37561 37567 37571 37573
37579 37589 37591 37607 37619 37633 37643 37649 37657 37663
37691 37693 37699 37717 37747 37781 37783 37799 37811 37813
37831 37847 37853 37861 37871 37879 37889 37897 37907 37951
37957 37963 37967 37987 37991 37993 37997 38011 38039 38047
38053 38069 38083 38113 38119 38149 38153 38167 38177 38183
38189 38197 38201 38219 38231 38237 38239 38261 38273 38281
38287 38299 38303 38317 38321 38327 38329 38333 38351 38371
38377 38393 38431 38447 38449 38453 38459 38461 38501 38543
38557 38561 38567 38569 38593 38603 38609 38611 38629 38639
38651 38653 38669 38671 38677 38693 38699 38707 38711 38713
38723 38729 38737 38747 38749 38767 38783 38791 38803 38821
38833 38839 38851 38861 38867 38873 38891 38903 38917 38921
38923 38933 38953 38959 38971 38977 38993 39019 39023 39041
39043 39047 39079 39089 39097 39103 39107 39113 39119 39133
39139 39157 39161 39163 39181 39191 39199 39209 39217 39227
39229 39233 39239 39241 39251 39293 39301 39313 39317 39323
39341 39343 39359 39367 39371 39373 39383 39397 39409 39419
39439 39443 39451 39461 39499 39503 39509 39511 39521 39541
39551 39563 39569 39581 39607 39619 39623 39631 39659 39667
39671 39679 39703 39709 39719 39727 39733 39749 39761 39769
39779 39791 39799 39821 39827 39829 39839 39841 39847 39857
39863 39869 39877 39883 39887 39901 39929 39937 39953 39971
39979 39983 39989 40009 40013 40031 40037 40039 40063 40087
40093 40099 40111 40123 40127 40129 40151 40153 40163 40169
40177 40189 40193 40213 40231 40237 40241 40253 40277 40283
40289 40343 40351 40357 40361 40387 40423 40427 40429 40433
40459 40471 40483 40487 40493 40499 40507 40519 40529 40531
40543 40559 40577 40583 40591 40597 40609 40627 40637 40639
40693 40697 40699 40709 40739 40751 40759 40763 40771 40787
40801 40813 40819 40823 40829 40841 40847 40849 40853 40867
40879 40883 40897 40903 40927 40933 40939 40949 40961 40973
40993 41011 41017 41023 41039 41047 41051 41057 41077 41081
41113 41117 41131 41141 41143 41149 41161 41177 41179 41183
41189 41201 41203 41213 41221 41227 41231 41233 41243 41257
41263 41269 41281 41299 41333 41341 41351 41357 41381 41387
41389 41399 41411 41413 41443 41453 41467 41479 41491 41507
41513 41519 41521 41539 41543 41549 41579 41593 41597 41603
41609 41611 41617 41621 41627 41641 41647 41651 41659 41669
41681 41687 41719 41729 41737 41759 41761 41771 41777 41801
41809 41813 41843 41849 41851 41863 41879 41887 41893 41897
41903 41911 41927 41941 41947 41953 41957 41959 41969 41981
41983 41999 42013 42017 42019 42023 42043 42061 42071 42073
42083 42089 42101 42131 42139 42157 42169 42179 42181 42187
42193 42197 42209 42221 42223 42227 42239 42257 42281 42283
42293 42299 42307 42323 42331 42337 42349 42359 42373 42379
42391 42397 42403 42407 42409 42433 42437 42443 42451 42457
42461 42463 42467 42473 42487 42491 42499 42509 42533 42557
42569 42571 42577 42589 42611 42641 42643 42649 42667 42677
42683 42689 42697 42701 42703 42709 42719 42727 42737 42743
42751 42767 42773 42787 42793 42797 42821 42829 42839 42841
42853 42859 42863 42899 42901 42923 42929 42937 42943 42953
42961 42967 42979 42989 43003 43013 43019 43037 43049 43051
43063 43067 43093 43103 43117 43133 43151 43159 43177 43189
43201 43207 43223 43237 43261 43271 43283 43291 43313 43319
43321 43331 43391 43397 43399 43403 43411 43427 43441 43451
43457 43481 43487 43499 43517 43541 43543 43573 43577 43579
43591 43597 43607 43609 43613 43627 43633 43649 43651 43661
43669 43691 43711 43717 43721 43753 43759 43777 43781 43783
43787 43789 43793 43801 43853 43867 43889 43891 43913 43933
43943 43951 43961 43963 43969 43973 43987 43991 43997 44017
44021 44027 44029 44041 44053 44059 44071 44087 44089 44101
44111 44119 44123 44129 44131 44159 44171 44179 44189 44201
44203 44207 44221 44249 44257 44263 44267 44269 44273 44279
44281 44293 44351 44357 44371 44381 44383 44389 44417 44449
44453 44483 44491 44497 44501 44507 44519 44531 44533 44537
44543 44549 44563 44579 44587 44617 44621 44623 44633 44641
44647 44651 44657 44683 44687 44699 44701 44711 44729 44741
44753 44771 44773 44777 44789 44797 44809 44819 44839 44843
44851 44867 44879 44887 44893 44909 44917 44927 44939 44953
44959 44963 44971 44983 44987 45007 45013 45053 45061 45077
45083 45119 45121 45127 45131 45137 45139 45161 45179 45181
45191 45197 45233 45247 45259 45263 45281 45289 45293 45307
45317 45319 45329 45337 45341 45343 45361 45377 45389 45403
45413 45427 45433 45439 45481 45491 45497 45503 45523 45533
45541 45553 45557 45569 45587 45589 45599 45613 45631 45641
45659 45667 45673 45677 45691 45697 45707 45737 45751 45757
45763 45767 45779 45817 45821 45823 45827 45833 45841 45853
45863 45869 45887 45893 45943 45949 45953 45959 45971 45979
45989 46021 46027 46049 46051 46061 46073 46091 46093 46099
46103 46133 46141 46147 46153 46171 46181 46183 46187 46199
46219 46229 46237 46261 46271 46273 46279 46301 46307 46309
46327 46337 46349 46351 46381 46399 46411 46439 46441 46447
46451 46457 46471 46477 46489 46499 46507 46511 46523 46549
46559 46567 46573 46589 46591 46601 46619 46633 46639 46643
46649 46663 46679 46681 46687 46691 46703 46723 46727 46747
46751 46757 46769 46771 46807 46811 46817 46819 46829 46831
46853 46861 46867 46877 46889 46901 46919 46933 46957 46993
46997 47017 47041 47051 47057 47059 47087 47093 47111 47119
47123 47129 47137 47143 47147 47149 47161 47189 47207 47221
47237 47251 47269 47279 47287 47293 47297 47303 47309 47317
47339 47351 47353 47363 47381 47387 47389 47407 47417 47419
47431 47441 47459 47491 47497 47501 47507 47513 47521 47527
47533 47543 47563 47569 47581 47591 47599 47609 47623 47629
47639 47653 47657 47659 47681 47699 47701 47711 47713 47717
47737 47741 47743 47777 47779 47791 47797 47807 47809 47819
47837 47843 47857 47869 47881 47903 47911 47917 47933 47939
47947 47951 47963 47969 47977 47981 48017 48023 48029 48049
48073 48079 48091 48109 48119 48121 48131 48157 48163 48179
48187 48193 48197 48221 48239 48247 48259 48271 48281 48299
48311 48313 48337 48341 48353 48371 48383 48397 48407 48409
48413 48437 48449 48463 48473 48479 48481 48487 48491 48497
48523 48527 48533 48539 48541 48563 48571 48589 48593 48611
48619 48623 48647 48649 48661 48673 48677 48679 48731 48733
48751 48757 48761 48767 48779 48781 48787 48799 48809 48817
48821 48823 48847 48857 48859 48869 48871 48883 48889 48907
48947 48953 48973 48989 48991 49003 49009 49019 49031 49033
49037 49043 49057 49069 49081 49103 49109 49117 49121 49123
49139 49157 49169 49171 49177 49193 49199 49201 49207 49211
49223 49253 49261 49277 49279 49297 49307 49331 49333 49339
49363 49367 49369 49391 49393 49409 49411 49417 49429 49433
49451 49459 49463 49477 49481 49499 49523 49529 49531 49537
49547 49549 49559 49597 49603 49613 49627 49633 49639 49663
49667 49669 49681 49697 49711 49727 49739 49741 49747 49757
49783 49787 49789 49801 49807 49811 49823 49831 49843 49853
49871 49877 49891 49919 49921 49927 49937 49939 49943 49957
49991 49993 49999 50021 50023 50033 50047 50051 50053 50069
50077 50087 50093 50101 50111 50119 50123 50129 50131 50147
50153 50159 50177 50207 50221 50227 50231 50261 50263 50273
50287 50291 50311 50321 50329 50333 50341 50359 50363 50377
50383 50387 50411 50417 50423 50441 50459 50461 50497 50503
50513 50527 50539 50543 50549 50551 50581 50587 50591 50593
50599 50627 50647 50651 50671 50683 50707 50723 50741 50753
50767 50773 50777 50789 50821 50833 50839 50849 50857 50867
50873 50891 50893 50909 50923 50929 50951 50957 50969 50971
50989 50993 51001 51031 51043 51047 51059 51061 51071 51109
51131 51133 51137 51151 51157 51169 51193 51197 51199 51203
51217 51229 51239 51241 51257 51263 51283 51287 51307 51329
51341 51343 51347 51349 51361 51383 51407 51413 51419 51421
51427 51431 51437 51439 51449 51461 51473 51479 51481 51487
51503 51511 51517 51521 51539 51551 51563 51577 51581 51593
51599 51607 51613 51631 51637 51647 51659 51673 51679 51683
51691 51713 51719 51721 51749 51767 51769 51787 51797 51803
51817 51827 51829 51839 51853 51859 51869 51871 51893 51899
51907 51913 51929 51941 51949 51971 51973 51977 51991 52009
52021 52027 52051 52057 52067 52069 52081 52103 52121 52127
52147 52153 52163 52177 52181 52183 52189 52201 52223 52237
52249 52253 52259 52267 52289 52291 52301 52313 52321 52361
52363 52369 52379 52387 52391 52433 52453 52457 52489 52501
52511 52517 52529 52541 52543 52553 52561 52567 52571 52579
52583 52609 52627 52631 52639 52667 52673 52691 52697 52709
52711 52721 52727 52733 52747 52757 52769 52783 52807 52813
52817 52837 52859 52861 52879 52883 52889 52901 52903 52919
52937 52951 52957 52963 52967 52973 52981 52999 53003 53017
53047 53051 53069 53077 53087 53089 53093 53101 53113 53117
53129 53147 53149 53161 53171 53173 53189 53197 53201 53231
53233 53239 53267 53269 53279 53281 53299 53309 53323 53327
53353 53359 53377 53381 53401 53407 53411 53419 53437 53441
53453 53479 53503 53507 53527 53549 53551 53569 53591 53593
53597 53609 53611 53617 53623 53629 53633 53639 53653 53657
53681 53693 53699 53717 53719 53731 53759 53773 53777 53783
53791 53813 53819 53831 53849 53857 53861 53881 53887 53891
53897 53899 53917 53923 53927 53939 53951 53959 53987 53993
54001 54011 54013 54037 54049 54059 54083 54091 54101 54121
54133 54139 54151 54163 54167 54181 54193 54217 54251 54269
54277 54287 54293 54311 54319 54323 54331 54347 54361 54367
54371 54377 54401 54403 54409 54413 54419 54421 54437 54443
54449 54469 54493 54497 54499 54503 54517 54521 54539 54541
54547 54559 54563 54577 54581 54583 54601 54617 54623 54629
54631 54647 54667 54673 54679 54709 54713 54721 54727 54751
54767 54773 54779 54787 54799 54829 54833 54851 54869 54877
54881 54907 54917 54919 54941 54949 54959 54973 54979 54983
55001 55009 55021 55049 55051 55057 55061 55073 55079 55103
55109 55117 55127 55147 55163 55171 55201 55207 55213 55217
55219 55229 55243 55249 55259 55291 55313 55331 55333 55337
55339 55343 55351 55373 55381 55399 55411 55439 55441 55457
55469 55487 55501 55511 55529 55541 55547 55579 55589 55603
55609 55619 55621 55631 55633 55639 55661 55663 55667 55673
55681 55691 55697 55711 55717 55721 55733 55763 55787 55793
55799 55807 55813 55817 55819 55823 55829 55837 55843 55849
55871 55889 55897 55901 55903 55921 55927 55931 55933 55949
55967 55987 55997 56003 56009 56039 56041 56053 56081 56087
56093 56099 56101 56113 56123 56131 56149 56167 56171 56179
56197 56207 56209 56237 56239 56249 56263 56267 56269 56299
56311 56333 56359 56369 56377 56383 56393 56401 56417 56431
56437 56443 56453 56467 56473 56477 56479 56489 56501 56503
56509 56519 56527 56531 56533 56543 56569 56591 56597 56599
56611 56629 56633 56659 56663 56671 56681 56687 56701 56711
56713 56731 56737 56747 56767 56773 56779 56783 56807 56809
56813 56821 56827 56843 56857 56873 56891 56893 56897 56909
56911 56921 56923 56929 56941 56951 56957 56963 56983 56989
56993 56999 57037 57041 57047 57059 57073 57077 57089 57097
57107 57119 57131 57139 57143 57149 57163 57173 57179 57191
57193 57203 57221 57223 57241 57251 57259 57269 57271 57283
57287 57301 57329 57331 57347 57349 57367 57373 57383 57389
57397 57413 57427 57457 57467 57487 57493 57503 57527 57529
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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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free downloadable college text book
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 projectBuilding 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, 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) free downloadable college text book on computer programming. |
This web site handcrafted on Macintosh 
computers using Tom Bender’s Tex-Edit Plus 
and served using FreeBSD 
.
†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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