(* You should be able to copy and paste this text file directly into Mathematica.  These notes will be interpreted as commented code. *)


(* ********* Here are the numbers from the problems ******** *)

problem1m  := 282006548310266500521595796103735803923312201934083188120704096090962921710642902266245001
problem1w  := 169171645092006578241300799123542803090460485666471678762142554544526241248418101659881815
problem1N  := 340424449758185542695947485837945311955875942240280273796932383801516352255982079556106287

problem2N  := 51300594275226581813688476659949247556215009836233
problem2e  := 65537
problem2c  := 42532231210824048250926791283200710247749142953927

problem3Nb := 49703407978872135768369150951737194603841663052986938247511157126794635921277619
problem3Nc := 48394585785126752760098222942433754518772506574482068079987934034981215730453293
problem3Nd := 37048466581842421945081537172098726013070671280095643279361407260434395186752267
problem3cb := 05905364385466286295586251025237668938472190855132358966957728964323606634400251
problem3cc := 21138220486961146446206617482811850561629767638994082201111978852676605086081807
problem3cd := 27157125477984404879431019780288127319483825029543848767280738662683083014939218

problem6N  := 488419441734583556321985415212612123740359939381088965700730231638206554681394177
problem6s2 := 364578471930898294925524638136447727960007605573204140075455802888652544203808336
problem6u12:= 419987940537002829673554859623446087647247049378701209589622515994832674140645748
problem6u22:= 270893145623915322344834242328268768371424519375297223857305039560421032101793802
problem6u32:= 001204179001250513038323769136188667129468312291612708387897338022926559640599640
problem6u42:= 295360259330799676568102779994887111797263481168605647699269117672956353312755331
problem6u1 := 368836285783665928691160226566669484193845816214794656578305054442600293140251910
problem6su2:= 061162076090849776429311938634702834494489117638106960807555056103441302535633013
problem6su3:= 304092941078945109049230333889649788448145691132625713366641946452437843830720793
problem6su4:= 174908257541270590422202403049766598633440061550219493518183063157021792026188460

problem7cN := 45737 * 54377
problem7d  := 1657960491


(* ******* Here is code for decoding a number string, for problems 2 and 3 ******* *)

NumbersToString[list_] := StringJoin[FromLetterNumber[list + 1]]

NumberStringList[n_] := 
 PadLeft[IntegerDigits[n], 2*Ceiling[Length[IntegerDigits[n]]/2]]

NumberListConvert[list_] := 
 Table[FromDigits[list[[2 i - 1 ;; 2 i]]], {i, 1, Length[list]/2}]

NumberToString2[n_] := 
 NumbersToString[NumberListConvert[NumberStringList[n]]]

(*This will convert an integer into its corresponding letter string,where a=00,b=01,...,z=25*)


NumberToString2[0704111114]

(*This should return the string "hello"*)



(* ********* Here are some useful commands for powers mod m ******** *)

Mod[100,6]

(* This will compute 100 modulo 6, yielding 2 *)


PrimeQ[201]

(* This will test whether 201 is prime, which it is not *)


FactorInteger[143]

(* This will factor the integer 143, which is 11 times 13 *)
(* Mathematica's factorization algorithm knows how to do trial division, Pollard p-1, Pollard rho, quadratic sieving, and elliptic curve factorization, and is typically fairly fast on numbers with 50 digits or fewer *)


PowerMod[2, 10, 99]

(* This will compute 2^10 modulo 99.  PowerMod is efficient and will use successive squaring *)
(* Mod will often also recognize if you give it a power and internally call PowerMod, but it is better to use PowerMod directly *)


PowerMod[2, -1, 99]

(* This will compute 2^(-1) modulo 99, which is to say, the multiplicative inverse of 2 modulo 99. *)


EulerPhi[40]

(* This will compute phi(40). *)


ChineseRemainder[{1,2,3},{4,5,9}]

(* This will compute the smallest positive integer solution to x = 1 mod 4, x = 2 mod 5, x = 3 mod 9 *)


CubeRoot[64]

(* This will compute the cube root of 64.  You can also obtain this as 64^(1/3), but fractional powers can sometimes behave unexpectedly because of how Mathematica handles complex numbers. *)