Merge pull request #211 from christianbender/master

primelib

Project Euler/Problem 01/sol1.py

@@ -5,8 +5,8 @@ we get 3,5,6 and 9. The sum of these multiples is 23.
 Find the sum of all the multiples of 3 or 5 below N.
 '''
 n = int(raw_input().strip())
-sum=0;
+sum=0
 for a in range(3,n):
     if(a%3==0 or a%5==0):
         sum+=a
-print sum;
+print sum

Project Euler/Problem 01/sol3.py

@@ -8,8 +8,8 @@ Find the sum of all the multiples of 3 or 5 below N.
 This solution is based on the pattern that the successive numbers in the series follow: 0+3,+2,+1,+3,+1,+2,+3.
 '''
 n = int(raw_input().strip())
-sum=0;
-num=0;
+sum=0
+num=0
 while(1):
     num+=3
     if(num>=n):
@@ -39,4 +39,4 @@ while(1):
     if(num>=n):
         break
     sum+=num
-print sum;
+print sum

Project Euler/Problem 02/sol1.py

@@ -8,7 +8,9 @@ e.g. for n=10, we have {2,8}, sum is 10.
 '''
 
 n = int(raw_input().strip())
-i=1; j=2; sum=0
+i=1
+j=2 
+sum=0
 while(j<=n):
     if((j&1)==0): #can also use (j%2==0)
         sum+=j

Project Euler/Problem 03/sol1.py

@@ -17,7 +17,7 @@ def isprime(no):
             return False
     return True
 
-max=0
+maxNumber = 0
 n=int(input())
 if(isprime(n)):
     print n
@@ -31,8 +31,8 @@ else:
         for i in range(3,n1,2):
             if(n%i==0):
                 if(isprime(n/i)):
-                    max=n/i
+                    maxNumber = n/i
                     break
                 elif(isprime(i)):
-                    max=i
-        print max
+                    maxNumber = i
+        print maxNumber

Project Euler/Problem 29/solution.py

@@ -0,0 +1,34 @@
+def main():
+    """ 
+    Consider all integer combinations of ab for 2 <= a <= 5 and 2 <= b <= 5:
+
+    22=4, 23=8, 24=16, 25=32
+    32=9, 33=27, 34=81, 35=243
+    42=16, 43=64, 44=256, 45=1024
+    52=25, 53=125, 54=625, 55=3125
+    If they are then placed in numerical order, with any repeats removed, we get the following sequence of 15 distinct terms:
+
+    4, 8, 9, 16, 25, 27, 32, 64, 81, 125, 243, 256, 625, 1024, 3125
+
+    How many distinct terms are in the sequence generated by ab for 2 <= a <= 100 and 2 <= b <= 100?
+    """
+
+    collectPowers = set()
+
+    currentPow = 0
+
+    N = 101     # maximum limit
+
+    for a in range(2,N):
+
+        for b in range (2,N):
+
+            currentPow = a**b   # calculates the current power
+            collectPowers.add(currentPow)   # adds the result to the set
+            
+
+    print "Number of terms ", len(collectPowers)
+
+
+if __name__ == '__main__':
+    main()

other/primelib.py

@@ -0,0 +1,605 @@
+# -*- coding: utf-8 -*-
+"""
+Created on Thu Oct  5 16:44:23 2017
+
+@author: Christian Bender
+
+This python library contains some useful functions to deal with
+prime numbers and whole numbers.
+
+Overview:
+
+isPrime(number)
+sieveEr(N)
+getPrimeNumbers(N)
+primeFactorization(number)
+greatestPrimeFactor(number)
+smallestPrimeFactor(number)
+getPrime(n)
+getPrimesBetween(pNumber1, pNumber2) 
+
+----
+
+isEven(number)
+isOdd(number)
+gcd(number1, number2)  // greatest common divisor
+kgV(number1, number2)  // least common multiple
+getDivisors(number)    // all divisors of 'number' inclusive 1, number
+isPerfectNumber(number)
+
+NEW-FUNCTIONS
+
+simplifyFraction(numerator, denominator)
+factorial (n) // n!
+fib (n) // calculate the n-th fibonacci term.
+
+-----
+
+goldbach(number)  // Goldbach's assumption
+
+"""
+
+def isPrime(number):
+    """
+        input: positive integer 'number'
+        returns true if 'number' is prime otherwise false.
+    """
+    import math # for function sqrt
+    
+    # precondition
+    assert isinstance(number,int) and (number >= 0) , \
+    "'number' must been an int and positive"
+    
+    status = True
+    
+    # 0 and 1 are none primes. 
+    if number <= 1:
+        status = False
+    
+    for divisor in range(2,int(round(math.sqrt(number)))+1):
+        
+        # if 'number' divisible by 'divisor' then sets 'status'
+        # of false and break up the loop. 
+        if number % divisor == 0:
+            status = False
+            break
+    
+    # precondition
+    assert isinstance(status,bool), "'status' must been from type bool"    
+    
+    return status
+
+# ------------------------------------------
+
+def sieveEr(N):
+    """
+        input: positive integer 'N' > 2
+        returns a list of prime numbers from 2 up to N.
+        
+        This function implements the algorithm called
+        sieve of erathostenes.         
+        
+    """
+    
+    # precondition
+    assert isinstance(N,int) and (N > 2), "'N' must been an int and > 2"
+    
+    # beginList: conatins all natural numbers from 2 upt to N
+    beginList = [x for x in range(2,N+1)]
+
+    ans = [] # this list will be returns.     
+    
+    # actual sieve of erathostenes
+    for i in range(len(beginList)):
+        
+        for j in range(i+1,len(beginList)):
+            
+            if (beginList[i] != 0) and \
+            (beginList[j] % beginList[i] == 0):
+                beginList[j] = 0
+    
+    # filters actual prime numbers.           
+    ans = [x for x in beginList if x != 0]
+    
+    # precondition
+    assert isinstance(ans,list), "'ans' must been from type list"    
+    
+    return ans
+    
+
+# --------------------------------
+
+def getPrimeNumbers(N):
+    """
+        input: positive integer 'N' > 2
+        returns a list of prime numbers from 2 up to N (inclusive)
+        This function is more efficient as function 'sieveEr(...)'
+    """    
+    
+    # precondition
+    assert isinstance(N,int) and (N > 2), "'N' must been an int and > 2"
+    
+    ans = []    
+    
+    # iterates over all numbers between 2 up to N+1 
+    # if a number is prime then appends to list 'ans'
+    for number in range(2,N+1):
+        
+        if isPrime(number):
+            
+            ans.append(number)
+   
+    # precondition
+    assert isinstance(ans,list), "'ans' must been from type list"
+        
+    return ans
+
+
+# -----------------------------------------
+    
+def primeFactorization(number):
+    """
+        input: positive integer 'number' 
+        returns a list of the prime number factors of 'number'
+    """
+
+    import math    # for function sqrt
+    
+    # precondition
+    assert isinstance(number,int) and number >= 0, \
+    "'number' must been an int and >= 0"
+    
+    ans = [] # this list will be returns of the function.
+
+    # potential prime number factors.
+
+    factor = 2    
+
+    quotient = number
+    
+    
+    if number == 0 or number == 1:
+        
+        ans.append(number)
+        
+    # if 'number' not prime then builds the prime factorization of 'number'    
+    elif not isPrime(number):
+    
+        while (quotient != 1):
+            
+            if isPrime(factor) and (quotient % factor == 0):
+                    ans.append(factor)
+                    quotient /= factor
+            else:
+                    factor += 1
+    
+    else:
+        ans.append(number)
+    
+    # precondition
+    assert isinstance(ans,list), "'ans' must been from type list"    
+    
+    return ans
+    
+
+# -----------------------------------------
+    
+def greatestPrimeFactor(number):
+    """
+        input: positive integer 'number' >= 0
+        returns the greatest prime number factor of 'number'
+    """
+    
+    # precondition
+    assert isinstance(number,int) and (number >= 0), \
+    "'number' bust been an int and >= 0"
+    
+    ans = 0    
+    
+    # prime factorization of 'number'
+    primeFactors = primeFactorization(number)
+
+    ans = max(primeFactors)     
+    
+    # precondition
+    assert isinstance(ans,int), "'ans' must been from type int"    
+    
+    return ans
+    
+
+# ----------------------------------------------
+    
+    
+def smallestPrimeFactor(number):
+    """
+        input: integer 'number' >= 0
+        returns the smallest prime number factor of 'number'
+    """
+    
+    # precondition
+    assert isinstance(number,int) and (number >= 0), \
+    "'number' bust been an int and >= 0"
+    
+    ans = 0    
+    
+    # prime factorization of 'number'
+    primeFactors = primeFactorization(number)
+    
+    ans = min(primeFactors)
+
+    # precondition
+    assert isinstance(ans,int), "'ans' must been from type int"    
+    
+    return ans
+    
+    
+# ----------------------
+    
+def isEven(number):
+    """
+        input: integer 'number'
+        returns true if 'number' is even, otherwise false.
+    """   
+
+    # precondition
+    assert isinstance(number, int), "'number' must been an int" 
+    assert isinstance(number % 2 == 0, bool), "compare bust been from type bool"
+    
+    return number % 2 == 0
+    
+# ------------------------
+    
+def isOdd(number):
+    """
+        input: integer 'number'
+        returns true if 'number' is odd, otherwise false.
+    """   
+
+    # precondition
+    assert isinstance(number, int), "'number' must been an int" 
+    assert isinstance(number % 2 != 0, bool), "compare bust been from type bool"
+    
+    return number % 2 != 0
+    
+# ------------------------
+    
+    
+def goldbach(number):
+    """
+        Goldbach's assumption
+        input: a even positive integer 'number' > 2
+        returns a list of two prime numbers whose sum is equal to 'number'
+    """
+    
+    # precondition
+    assert isinstance(number,int) and (number > 2) and isEven(number), \
+    "'number' must been an int, even and > 2"
+    
+    ans = [] # this list will returned
+    
+    # creates a list of prime numbers between 2 up to 'number'
+    primeNumbers = getPrimeNumbers(number)
+    lenPN = len(primeNumbers)    
+
+    # run variable for while-loops.
+    i = 0
+    j = 1
+    
+    # exit variable. for break up the loops
+    loop = True
+    
+    while (i < lenPN and loop):
+        
+        j = i+1;
+        
+        
+        while (j < lenPN and loop):
+            
+            if primeNumbers[i] + primeNumbers[j] == number:
+                loop = False
+                ans.append(primeNumbers[i])
+                ans.append(primeNumbers[j])
+                
+            j += 1;
+            
+            
+        i += 1
+        
+    # precondition
+    assert isinstance(ans,list) and (len(ans) == 2) and \
+    (ans[0] + ans[1] == number) and isPrime(ans[0]) and isPrime(ans[1]), \
+    "'ans' must contains two primes. And sum of elements must been eq 'number'"
+        
+    return ans
+    
+# ----------------------------------------------
+
+def gcd(number1,number2):
+    """
+        Greatest common divisor
+        input: two positive integer 'number1' and 'number2'
+        returns the greatest common divisor of 'number1' and 'number2'
+    """
+    
+    # precondition
+    assert isinstance(number1,int) and isinstance(number2,int) \
+    and (number1 >= 0) and (number2 >= 0), \
+    "'number1' and 'number2' must been positive integer."
+
+    rest = 0    
+    
+    while number2 != 0:
+        
+        rest = number1 % number2
+        number1 = number2
+        number2 = rest
+
+    # precondition
+    assert isinstance(number1,int) and (number1 >= 0), \
+    "'number' must been from type int and positive"    
+    
+    return number1
+    
+# ----------------------------------------------------
+    
+def kgV(number1, number2):
+    """
+        Least common multiple
+        input: two positive integer 'number1' and 'number2'
+        returns the least common multiple of 'number1' and 'number2'
+    """
+    
+    # precondition
+    assert isinstance(number1,int) and isinstance(number2,int) \
+    and (number1 >= 1) and (number2 >= 1), \
+    "'number1' and 'number2' must been positive integer."
+    
+    ans = 1 # actual answer that will be return.
+     
+    # for kgV (x,1)
+    if number1 > 1 and number2 > 1:
+        
+        # builds the prime factorization of 'number1' and 'number2'
+        primeFac1 = primeFactorization(number1)
+        primeFac2 = primeFactorization(number2)
+        
+    elif number1 == 1 or number2 == 1:
+        
+        primeFac1 = []
+        primeFac2 = []
+        ans = max(number1,number2)
+    
+    count1 = 0
+    count2 = 0
+     
+    done = [] # captured numbers int both 'primeFac1' and 'primeFac2'
+    
+    # iterates through primeFac1
+    for n in primeFac1:
+        
+        if n not in done:
+        
+            if n in primeFac2:
+            
+                count1 = primeFac1.count(n)
+                count2 = primeFac2.count(n)
+            
+                for i in range(max(count1,count2)):
+                    ans *= n
+        
+            else:
+                
+                count1 = primeFac1.count(n)
+                
+                for i in range(count1):
+                    ans *= n
+                    
+            done.append(n)
+    
+    # iterates through primeFac2
+    for n in primeFac2:
+        
+        if n not in done:
+            
+            count2 = primeFac2.count(n)
+            
+            for i in range(count2):
+                ans *= n
+                    
+            done.append(n)
+    
+    # precondition
+    assert isinstance(ans,int) and (ans >= 0), \
+    "'ans' must been from type int and positive"    
+    
+    return ans
+    
+# ----------------------------------
+    
+def getPrime(n):
+    """
+        Gets the n-th prime number.
+        input: positive integer 'n' >= 0
+        returns the n-th prime number, beginning at index 0
+    """
+    
+    # precondition
+    assert isinstance(n,int) and (n >= 0), "'number' must been a positive int"
+    
+    index = 0
+    ans = 2 # this variable holds the answer
+    
+    while index < n:
+        
+        index += 1
+        
+        ans += 1   # counts to the next number     
+        
+        # if ans not prime then
+        # runs to the next prime number. 
+        while not isPrime(ans):
+            ans += 1
+    
+    # precondition
+    assert isinstance(ans,int) and isPrime(ans), \
+    "'ans' must been a prime number and from type int"    
+    
+    return ans
+    
+# ---------------------------------------------------
+    
+def getPrimesBetween(pNumber1, pNumber2):
+    """
+        input: prime numbers 'pNumber1' and 'pNumber2'
+                pNumber1 < pNumber2
+        returns a list of all prime numbers between 'pNumber1' (exclusiv)
+                and 'pNumber2' (exclusiv) 
+    """
+    
+    # precondition
+    assert isPrime(pNumber1) and isPrime(pNumber2) and (pNumber1 < pNumber2), \
+    "The arguments must been prime numbers and 'pNumber1' < 'pNumber2'"
+    
+    number = pNumber1 + 1 # jump to the next number
+    
+    ans = [] # this list will be returns.
+    
+    # if number is not prime then
+    # fetch the next prime number. 
+    while not isPrime(number):
+        number += 1
+    
+    while number < pNumber2:
+        
+        ans.append(number)
+        
+        number += 1
+        
+        # fetch the next prime number. 
+        while not isPrime(number):
+            number += 1
+            
+    # precondition
+    assert isinstance(ans,list) and ans[0] != pNumber1 \
+    and ans[len(ans)-1] != pNumber2, \
+    "'ans' must been a list without the arguments"
+            
+    # 'ans' contains not 'pNumber1' and 'pNumber2' !
+    return ans
+    
+# ----------------------------------------------------
+
+def getDivisors(n):
+    """
+        input: positive integer 'n' >= 1
+        returns all divisors of n (inclusive 1 and 'n')
+    """
+    
+    # precondition
+    assert isinstance(n,int) and (n >= 1), "'n' must been int and >= 1"
+
+    from math import sqrt        
+        
+    ans = [] # will be returned.
+    
+    for divisor in range(1,n+1):
+        
+        if n % divisor == 0:
+            ans.append(divisor)
+       
+       
+    #precondition
+    assert ans[0] == 1 and ans[len(ans)-1] == n, \
+    "Error in function getDivisiors(...)"
+    
+    
+    return ans
+
+
+# ----------------------------------------------------
+
+
+def isPerfectNumber(number):
+    """
+        input: positive integer 'number' > 1
+        returns true if 'number' is a perfect number otherwise false.
+    """
+    
+    # precondition
+    assert isinstance(number,int) and (number > 1), \
+    "'number' must been an int and >= 1"
+    
+    divisors = getDivisors(number)
+ 
+    # precondition
+    assert isinstance(divisors,list) and(divisors[0] == 1) and  \
+    (divisors[len(divisors)-1] == number), \
+    "Error in help-function getDivisiors(...)"
+    
+    # summed all divisors up to 'number' (exclusive), hence [:-1]
+    return sum(divisors[:-1]) == number
+
+# ------------------------------------------------------------
+
+def simplifyFraction(numerator, denominator):
+    """
+        input: two integer 'numerator' and 'denominator'
+        assumes: 'denominator' != 0
+        returns: a tuple with simplify numerator and denominator.
+    """      
+    
+    # precondition
+    assert isinstance(numerator, int) and isinstance(denominator,int) \
+    and (denominator != 0), \
+    "The arguments must been from type int and 'denominator' != 0"
+    
+    # build the greatest common divisor of numerator and denominator.
+    gcdOfFraction = gcd(abs(numerator), abs(denominator))
+
+    # precondition
+    assert isinstance(gcdOfFraction, int) and (numerator % gcdOfFraction == 0) \
+    and (denominator % gcdOfFraction == 0), \
+    "Error in function gcd(...,...)"
+    
+    return (numerator // gcdOfFraction, denominator // gcdOfFraction)
+    
+# -----------------------------------------------------------------
+    
+def factorial(n):
+    """
+        input: positive integer 'n'
+        returns the factorial of 'n' (n!)
+    """
+    
+    # precondition
+    assert isinstance(n,int) and (n >= 0), "'n' must been a int and >= 0"
+    
+    ans = 1 # this will be return.
+    
+    for factor in range(1,n+1):
+        ans *= factor
+    
+    return ans
+    
+# -------------------------------------------------------------------
+    
+def fib(n):
+    """
+        input: positive integer 'n'
+        returns the n-th fibonacci term , indexing by 0
+    """  
+    
+    # precondition
+    assert isinstance(n, int) and (n >= 0), "'n' must been an int and >= 0"
+    
+    tmp = 0
+    fib1 = 1
+    ans = 1 # this will be return
+    
+    for i in range(n-1):
+        
+        tmp = ans
+        ans += fib1
+        fib1 = tmp
+        
+    return ans