At the unix prompt, type "python" (and hit return) and you'll get the python prompt: >>> To _stop_ python and return to the unix prompt, type "Control-D" at the python prompt. If python gets stuck in a loop or in a too-long process, "Control-C" interrupts it and returns you to the prompt. In its simplest use, the python prompt gives a read-evaluate-print loop, which has many built-in functions useful for basic computational number theory. Addition is +, multiplication is *, division is /, substraction is -, use of parentheses is as usual. A slightly less standard use is ** for exponentiation, so, for example, 2**5=32. The python prompt's loop can be used as a simple calculator, by entering an expression and hitting "return" to evaluate it: >>> 2*3 6 >>> 4 * 4 16 >>> 2**10 1024 >>> (3+5) * (1+2) 24 >>> Whitespace between operators doesn't matter. An unusual feature of python is that it has built-in infinite-precision integers, of arbitrary size. These are denoted by appending an "L" to the numeral. If the numbers get too large for ordinary integers, you'll get an overflow error: >>> 2 ** 100 Traceback (innermost last): File "", line 1, in ? OverflowError: integer pow() >>> But if you use "long" integers it'll be fine: >>> 2L ** 100 1267650600228229401496703205376L >>> Of course, you can still ask the machine to do tedious and stupid things, such as >>> 2L ** 1000000 ## If you do this, for example, the machine will appear to hang. If ## you realize that you've made a stupid request, you _can_ stop it ## in the middle without harming anything: do Control-C, and you'll ## be returned to the prompt. For use in crypto and other computational number theory, there is a built-in function which does _modular_exponentiation_: to compute x^e % m (x to the e power reduced modulo m) do _not_ do >>> (x**e) % m ## bad! but instead do >>> pow(x,e,m) ## good The latter uses a smart algorithm. So, for example, ridiculously large exponents are ok (as long as the "L" suffix is used): >>> pow(2L, 10L**100+1, 10L**100+1) 6660888126245688699182136543775795641628347778449324704651411345556300289268422357122875536174966145L takes an imperceptible amount of time to evaluate. Very good. ################################################################# ################################################################# There are built-in mathematical functions, too, but preparation for them requires _importing_ the "math module": do >>> import math just once at the beginning of a python session, to import the math module. Thereafter, the following functions and constants (among others) are available: math.log10( ) ## log base 10 math.pi ## 3.14159265359 math.sqrt( ) ## square root, in floating-point: watch out! math.acos( ) ## inverse cosine math.sin( ) ## sine function, in radians math.atan2(,) ## = arctan( firstargument / secondargument ) math.exp( ) ## exponentiation base e math.asin( ) ## arcsine math.floor( ) ## greatest integer less-than-or-equal-to: the floor function math.fabs( ) ## least integer greater-than-or-equal-to: the ceiling function math.log( ) ## natural log math.e ## 2.71828182846 math.atan( ) ## arctangent The usage as set up here requires the prefix "math", as in >>> math.log10(2) 0.301029995664 >>> On hazard is that these functions only keep a fixed number of decimal places of "precision", so ###################################################################### To _replay_ and _edit_ the last command, a nice subset of EMACS text-editing capabilities is available: among other things, a basic set of commands is ("C-x" is an abbreviation for "Control-x") ("M-x" is an abbreviation for "Meta-x") C-f ;; move forward by one character C-b ;; move backward by one character M-f ;; move forward by one word M-b ;; move backward by one word C-p ;; recall previous command (can be iterated further!) C-n ;; move to next command C-a ;; move to beginning of line C-e ;; move to end of line C-d ;; delete next character M-d ;; delete next word DEL ;; delete previous character M-DEL ;; delete previous word C-k ;; delete from point to end of line The arrow keys usually work, also. ###################################################################### A related organizational tool (as well as actualy programming capability) is the possibility of _named_variables_. This can be done at the python command line loop, using "=" to assign values to variables. >>> n = 1001 >>> n 1001 >>> n = n+1 >>> n 1002 >>> This allows more coherent treatment of complicated expressions: >>> n =1001 >>> n 1001 >>> n = n+1 >>> n 1002 >>> n = 1000001L >>> n 1000001L >>> a = pow(2,n,n) >>> a 210991L >>> ###################################################################### "While loops" are easy to write: >>> i=1 >>> while i<10: ## hit return at end of line ... print `i` ## hit tab at beginning of line, return at end ... i = i+1 ## hit tab at beginning of line, return at end ... ## hit return to close up the block 1 2 3 4 5 6 7 8 9 >>> Note that the backquotes convert _numbers_ to _strings_ Note that braces are _not_ used. Rather, code blocks are indicated by _indentation_. Thus, every line in the "while block" must be indented, conventionally by a tab. Sub-blocks have their own further indentation: >>> i = 1 ## hit return >>> j = 1 ## hit return >>> while i<5: ## hit return ... while j <= i: ## tab at beginning, return at end of line ... print `i+j` ## _two_ tabs at beginning, return at end ... j = j+1 ## _two_ tabs at beginning, return at end ... i = i+1 ## _one_ tab at beginning, return at end ... ## just a return, to close the outer block 2 4 6 8 >>> ###################################################################### "For loops" are similar, and also "if/else" statements are reasonable: >>> n = 1000001 >>> n % 3 2 >>> n = 1000001 >>> for i in range(1,1001): ... if n % i == 0: ... print `i` ... 1 101 >>> The syntax of the "else" clause is: Note that the _test_ for equality (as opposed to _assignment_ of variable values) is a _double_ equal sign. The "range" thingy tells the variable i to assume all integer values including the bottom but excluding the top value. It has an optional argument that tells how much to increment it (the default is 1): >>> for i in range(1,10,3): ... print `i` ... 1 4 7 >>> If the range is rather large, it may be more economical to use a variant construct "xrange", rather than plain "range", which does not create the whole list in memory at one time: >>> for i in xrange(1,100000): ... if i== 12345: ... print `i` ... elif i== 1234501: ... print "Hello" ... else: ... pass 12345 'Hello' >>> This example also showed how to print a string. It also showed how to have a "no-operation" occur: the keyword is "pass" ###################################################################### It is easy to define simple functions to use in the python command-line loop, using "def" and "return" keywords: >>> def f(x): ## hit return ... return x*x*x-3*x*x+17*x-1 ## hit tab at beginning of line, return at end ... ## hit return to end definition block >>> f(1) 14 >>> f(0) -1 >>> f(-1) -22 >>> f(2) 29 >>> Somewhat more complicated things can also be set up in code blocks, using "if/else" blocks and "for" or "while" loops: >>> def f(n): ... if n == 1: ... return 1 ... elif n==2: ... return 2 ... else: ... return 3 ... >>> f(4) 3 >>> Variable names and function names can be long and descriptive if you want: >>> def primality_test(n): ... bound = int( math.sqrt(n) ) ... for i in range(3,bound+1,2): ... if n % i == 0: ... return `i` + " divides " + `n` ... return `n` + " is prime " ... >>> primality_test(101) '101 is prime ' >>> primality_test(105) '3 divides 105' >>> Note that the "+" operator also concatenates _strings_, which can be delimited by double-quotes. ###################################################################### _List_ syntax is reasonable: >>> mylist = [1,2,3] >>> mylist [1, 2, 3] >>> mylist.append(101) >>> mylist [1, 2, 3, 101] >>> mylist.append(123) >>> mylist [1, 2, 3, 101, 123] >>> mylist Lists can be used in _for_loops_: >>> for i in mylist: ... print `i` ... 1 2 3 101 123 >>> Here's the time to point out a variant form of _print_, which doesn't insert a newline after each item, but only a space: >>> for i in mylist: ... print `i`, ... 1 2 3 101 123 >>> That is, the comma after the `i` causes the "print" to only insert a space rather than newline. The length of a list is accessible by the "len" function, which also can compute the length of a _string_: >>> len(mylist) 5 >>> len("hello") 5 >>> The various elements of a list are accessible by typical index notation, with indices beginning at 0: >>> mylist[2] 3 >>> mylist[0] 1 >>> _Slices_ of lists use a funny but reasonable notation >>> mylist[0:3] [1, 2, 3] >>> mylist[:-1] [1, 2, 3, 101] >>> mylist[:-2] [1, 2, 3] >>> mylist[0:-2] [1, 2, 3] >>> mylist[1:-2] [2, 3] >>> mylist[2:-2] [3] >>> mylist[3:-2] [] >>> myslice = mylist[3:] >>> myslice [101, 123] >>> ###################################################################### When you get tired of retyping at the command-line loop, you can (of course) put python code into files and _either_ "run" such files on their own, _or_ use them as "modules" from the python command-line loop itself. To load into the command loop files defining functions/subroutines/methods, these files should end with ".py", like "myfile.py", _and_ should either be in your current directory (in a unix sense), or in your python module search path. (At the unix prompt do "echo $PYTHONPATH" to see the value of this environmental variable.) Suppose that myfile.py contains just the two lines (ended by a newline!) def f(n): return n+1 (Assuming that the python interpreter can find the files, because they're in the search path), to load myfile.py into the python loop, do >>> import myfile Note that the suffix is not included. Then functions defined in myfile can be called by >>> myfile.f(5) 6 If the definition of myfile.f is changed in myfile.py and you want to "reload", the syntax is >>> reload(myfile) To load functions/methods in a manner so that the prefix "myfile." is not needed, use the syntax >>> from myfile import f >>> f(11) 12 >>> Such Python modules can include auxiliary functions called internally. Such internal use does not require use of the prefix naming the module. For example, myfile.py might contain the 3 lines (with newline at the end) import math def mysqrt(n): return math.sqrt(n) Then >>> reload(myfile) >>> myfile.mysqrt(16) 4.0 >>>

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This work by Paul Garrett is licensed under a Creative Commons Attribution 3.0 Unported License. ... [

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The University of Minnesota explicitly requires that I state that