Number Theory Integer Types by Michael Carter Carl Friedrich Gauss was known as one of the top ten greatest mathematicians. He was known as the prince of mathematics. Gauss is quoted calling mathematics the "queen of the sciences" and calling number theory the "queen of mathematics. is called "higher arithmetic." Number Theory is the study of whole numbers and sometimes spills over to rational numbers. The whole numbers that is the most fascinating are the primes. We know very little about the primes. Because we know so little about the primes we can make encryption for banks and computer security systems. We also have another fascinating number that we know more about, the composite numbers. The composite numbers are composed of There are approximately 10 fields within Number Theory: Elementary, Analytic, Algebraic, Geometry, Combinatorial, Computational, Arithmetic Algebraic, Arithmetic Topology, Arithmetic Dynamics, and Modular Form. All these fields can take advantage of this maple package. This Maple package is called inttypes, which is short for Integer Types. The inttypes package provides over 160 whole number types for Number Theorists to explore numbers. This package may be used as a tool for an academic course, Maple programming, or for research. It is also fun to play with these types for exploration and developing intuition about numbers. In addition, to these number theory integer types, this package also offers three very important functions: ithcomposite, ithprimorial, and printtypes. Maple have a function called, ithprime; which is used to find a particular prime based on its order. This inttypes package provides a function that gives a particular composite number based on its order, called ithcomposite. There is also an ithprimorial function that provides a particular primordial number based on its order. Finally, we have a function that will search among all the types of this package for a number, 1729, is a good test for printtypes (see below). The number, 3, is also a good test for the printtypes function. On the Number Theory Types Test: Just change the integer on each of the for-loops for larger tests when testing the number theory types. All types return a Boolean value of either true or false. It does not return a number. Note in the testing of the types below we can give input but the outpus is true or false. The input integer is not the output, hence, do not get confused and begin believing that the input is the output. Finally, there is an official website for referencing and testing the http://oeis.org/ I hope you enjoy this package as much as I enjoyed creating it for you. Cheers. ============================================================== =============================== Application Setup Instructions ============================================================= This package was designed on Maple 15. It may work on earlier versions of Maple, however, you would need to make modifications in the maple.ini file to make this happen. If you have not done so already, extract all the files from the file inttypes.zip to the "C:\inttypes" folder. These instructions assume that this location is "C:\inttypes" Place the maple.ini file in the "C:\Program Files\Maple 15\Users folder" Maple will automatically find this package when you load it using the "with" statement. If you do not have permission to place files in the "C:\Program Files\Maple 15\Users" folder then see "Alternate Setup Instructions." Start Maple. type with(inttypes); Alternate Setup Instructions ======================================== If you do not have permission to place files in the "C:\Program Files\Maple 15\Users" folder then To do this inside a worksheet, at the prompt, enter the commands: intlibname := "C:/inttypes": libname := intlibname, libname: with(inttypes); ============================================================== ======================================================== # Number Theory Type Type's Name == =================== ====================== #1) compositenum "Composite Number" #2) prime "Prime Number" (internal to maple) #3) semiprime "Semiprime" #4) almostprime3 "3-Almost Prime" #5) almostprime4 "4-Almost Prime" #6) almostprime5 "5-Almost Prime" #7) almostprime6 "6-Almost Prime" #8) almostprime7 "7-Almost Prime" #9) almostprime8 "8-Almost Prime" #10) almostprime9 "9-Almost Prime" #11) almostprime10 "10-Almost Prime" #12) almostprime11 "11-Almost Prime" #13) almostprime12 "12-Almost Prime" #14) almostprime13 "13-Almost Prime" #15) almostprime14 "14-Almost Prime" #16) almostprime15 "15-Almost Prime" #17) almostprime16 "16-Almost Prime" #18) almostprime17 "17-Almost Prime" #19) almostprime18 "18-Almost Prime" #20) almostprime19 "19-Almost Prime" #21) almostprime20 "20-Almost Prime" #22) perfectnum "Perfect Number" #23) mersennenum "Mersenne Number" #24) mersenneexpprime "Mersenne Exponent Prime" #25) mersenneprime "Mersenne Prime" #26) doublemersenneprime "Double Mersenne Prime" #27) fibonaccinum "Fibonacci Number" #28) fibonacciprime "Fibonacci Prime" #29) permutableprime "Permutable Prime" #30) lessertwinprime "Lesser Twin Prime" #31) greatertwinprime "Greater Twin Prime" #32) isolatedprime "Isolated Prime" #33) lessercousinprime "Lesser Cousin Prime" #34) greatercousinprime "Greater Cousin Prime" #35) lessersexyprime "Lesser Sexy Prime" #36) greatersexyprime "Greater Sexy Prime" #37) eisensteinprime "Eisenstein Prime" #38) gaussianprime "Gaussian Prime" #39) repdigitnum "Rep Digit Number" #40) repunitprime "Rep Unit Prime" #41) woodallnum "Woodall Number" #42) woodallprime "Woodall Prime" #43) palindromicnum "Palindromic Number" #44) beautifulnum "Beautiful Number" #45) squarenum "Square Number" #46) triangularnum "Triangular Number" #47) oblongnum "Oblong Number" #48) hexagonalnum "Hexagonal Number" #49) tetrahedralnum "Tetrahedral Number" #50) decagonalnum "Decagonal Number" #51) abundantnum "Abundant Number" #52) aspiringnum "Aspiring Number" #53) sociablenum "Sociable Number" #54) squarefreenum "Square-free Number" #55) luckynum "Lucky Number" #56) luckyprime "Lucky Prime" #57) pentagonalnum "Pentagonal Number" #58) amicablenum "Amicable Number" #59) deficientnum "Deficient Number" #60) happynum "Happy Number" #61) happyprime "Happy Prime" #62) unhappynum "Unhappy Number" #63) unhappyprime "Unhappy Prime" #64) pseudoperfectnum "Pseudo-perfect Number" #65) automorphicnum "Automorphic Number" #66) duffiniannum "Duffinian Number" #67) cuban1prime "Cuban 1 Prime" #68) cuban2prime "Cuban 2 Prime" #69) prothnum "Proth Number" #70) prothprime "Proth Prime" #71) wilsonprime "Wilson Prime" #72) bellnum "Bell Number" #73) bellprime "Bell Prime" #74) carolnum "Carol Number" #75) carolprime "Carol Prime" #76) cdecagonalnum "Centered Decagonal Number" #77) cdecagonalprime "Centered Decagonal Prime" #78) cheptagonalnum "Centered Heptagonal Number" #79) cheptagonalprime "Centered Heptagonal Prime" #80) csquarenum "Centered Square Number" #81) csquareprime "Centered Square Prime" #82) ctriangularnum "Centered Triangular Number" #83) ctriangularprime "Centered Triangular Prime" #84) chenprime "Chen Prime" #85) superprime "Super Prime" #86) supersingularprime "Super Singular Prime" #87) cullennum "Cullen Number" #88) cullenprime "Cullen Prime" #89) circularprime "Circular Prime" #90) emirps "Reverse Primes (emirps)" #91) primorialnum "Primorial Number" #92) primorialprime "Primorial Prime" #93) euclidnum "Euclid Number" #94) euclidprime "Euclid Prime" #95) evenprime "Even Prime" #96) oddprime "Odd Prime" #97) evennum "Even Number" #98) oddnum "Odd Number" #99) fermatnum "Fermat Number" #100) fermatprime "Fermat Prime" #101) genocchiprime "Genochi Prime" #102) goodprime "Good Prime" #103) kyneanum "Kynea Number" #104) kyneaprime "Kynea Prime" #105) lefttruncatableprime "Left Truncatable Prime" #106) righttruncatableprime "Right Truncatable Prime" #107) twosidedprime "Two Sided Prime" #108) lucasnum "Lucas Number" #109) lucasprime "Lucas Prime" #110) minimalprime "Minimal Prime" #111) padovannum "Padovan Number" #112) padovanprime "Padovan Prime" #113) partitionnum "Partition Number" #114) partitionprime "Partition Prime" #115) pellnum "Pell Number" #116) pellprime "Pell Prime" #117) perrinnum "Perrin Number" #118) perrinprime "Perrin Prime" #119) pierpontprime "Pierpont Prime" #120) pythagoreanprime "Pythagorean Prime" #121) binaryquadraticprime "Binary Quadratic Prime" #122) quartanprime "Quartan Prime" #123) residueclass6nplus1prime "Residue Class 6n + 1 Prime" #124) residueclass6nplus5prime "Residue Class 6n + 5 Prime" #125) residueclass8nplus1prime "Residue Class 8n + 1 Prime" #126) residueclass8nplus3prime "Residue Class 8n + 3 Prime" #127) residueclass8nplus5prime "Residue Class 8n + 5 Prime" #128) residueclass8nplus7prime "Residue Class 8n + 7 Prime" #129) residueclass10nplus1prime "Residue Class 10n + 1 Prime" #130) residueclass10nplus3prime "Residue Class 10n + 3 Prime" #131) residueclass10nplus7prime "Residue Class 10n + 7 Prime" #132) residueclass10nplus9prime "Residue Class 10n + 9 Prime" #133) safeprime "Safe Prime" #134) sophiegermainprime "Sophie Germain Prime" #135) starprime "Star Prime" #136) thabitprime "Thabit Prime" #137) wagstaffformprime "Wagstaff Form of (2p + 1) / 3 Prime" #138) wagstaffvalueprime "Wagstaff Value p Prime" #139) wieferichbase2prime "Wieferich base 2 prime" #140) wieferichbase3prime "Wieferich base 3 prime" #141) wieferichbase5prime "Wieferich base 5 prime" #142) wieferichbase6prime "Wieferich base 6 prime" #143) wieferichbase7prime "Wieferich base 7 prime" #144) wieferichbase10prime