def primeAndTwiceSq(n):
for sumPart in sieveOfEratosthenes(n):
twiceSquare = n- sumPart
if twiceSquare%2==0:
if sqrt(twiceSquare/2)==int(sqrt(twiceSquare/2)):
return [sumPart,2,twiceSquare/2]
return[n,-1,-1]
## SOLVED # How many numbers below fifty million can be expressed as the # sum of a prime square, prime cube, and prime fourth power? # http://projecteuler.net/problem=87 from time import clock from samsPrimeTools import sieveOfEratosthenes # start timer st=clock() # sieve for primes up to sqrt(50m) primes=sieveOfEratosthenes(7072) # remove zeroes for z in range(primes.count(0)): primes.remove(0) # set object will not contain # duplicated entries S=set() # limit is fourth root of 50m # this won't BE 85 since the # zeroes have been removed for f in primes[:85]: if f<85:
# a permutation of 79180.
# Find the value of n, 1 < n < 107, for which phi(n) is a
# permutation of n and the ratio n/phi(n) produces a minimum.
from samsPrimeTools import sieveOfEratosthenes
def ispermutation(a,b):
if sorted(str(a))==sorted(str(b)): return True
else: return False
print 'Sieving'
primes=sieveOfEratosthenes(4000)
for i in range(primes.count(0)):
primes.remove(0)
print 'Finding totients'
i=0
minratio=2
for p1 in primes:
i+=1
for p2 in primes[i:]:
n=p1*p2
if n > 10**7:break
phi=(p1-1)*(p2-1)
if n==87109:print n,phi
# irrational. The decimal expansion of such square
# roots is infinite without any repeating pattern at
# all.
#
# The square root of two is 1.41421356237309504880..
# and the digital sum of the first one hundred
# decimal digits is 475.
#
# For the first one hundred natural numbers, find
# the total of the digital sums of the first one
# hundred decimal digits for all the irrational
# square roots.
from samsPrimeTools import sieveOfEratosthenes
from decimal import *
getcontext().prec = 100
primes=sieveOfEratosthenes(100)
t=0
for p in primes:
if p!=0:
s=str(Decimal(p).sqrt())
s=s[2:]
for d in s:
t+=int(d)
print t
# 33 = 3**2 + 2**3 + 2**4
# 49 = 5**2 + 2**3 + 2**4
# 47 = 2**2 + 3**3 + 2**4
# How many numbers below fifty million can be expressed as the sum
# of a prime square, prime cube, and prime fourth power?
# ==============================================================
from samsPrimeTools import sieveOfEratosthenes
#upper limit is the square root of fifty million
upper= int(5e7**(0.5))+1
print 'Sieving primes'
primes=sieveOfEratosthenes(upper)
print 'Removing zeros'
for z in range(primes.count(0)):
primes.remove(0)
print primes
print 'Searching'
s=set()
for a in primes:
for b in primes:
for c in primes:
t=(a**2) + (b**3) + (c**4)
if t<50:s.add(t)
