46 lines
1.3 KiB
Python
46 lines
1.3 KiB
Python
#
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#Special Pythagorean triplet
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#Problem 9
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#
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#A Pythagorean triplet is a set of three natural numbers, a < b < c, for which,
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#
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#a^2 + b^2 = c^2
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#For example, 3^2 + 4^2 = 9 + 16 = 25 = 5^2.
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#
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#There exists exactly one Pythagorean triplet for which a + b + c = 1000.
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#Find the product abc.
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#
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#Solution:
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#
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# a^2+b^2=c^2=(1000-a-b)^2= 1000^2-2000(a+b)+(a+b)^2=
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# = 1000^2-2000(a+b)+a^2+2ab+b^2 =>
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# => 0=1000^2-2000(a+b)+2ab =>
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# => 500000=1000(a+b)-ab
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# => 500=a+b - ab/1000
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# => ab is dividable to 1000=2*2*2*5*5*5
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# => a dividable at least to 5 and b to 2
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#
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#for a in range(5,1000,5):
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# for b in range(2,1000-a,2):
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# i+=1
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# if (a**2+b**2) == (1000-a-b)**2:
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# print( a, b, 1000-a-b, '->', a**2, '+', b**2, '=', a**2+b**2 )
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# print( ' while (1000-a-b)**2 =', (1000-a-b)**2 )
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# print( 'product =', a*b*(1000-a-b) )
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# print( 'product =', (a+b)*(5*10**5+10**3) - (a+b)**2 - 5*10**8 )
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#
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# a better solution
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# => 500000=1000(a+b)-ab
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# => (500000-1000a)/(1000-a)=b
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# => 1000(500-a)/(1000-a)=b
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for a in range(1,500):
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r=1000*(500-a)
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q=1000-a
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b=r//q
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if (a**2+b**2) == (1000-a-b)**2:
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print( a, b, 1000-a-b, '->', a**2, '+', b**2, '=', a**2+b**2 )
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print( ' while (1000-a-b)**2 =', (1000-a-b)**2 )
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print( 'product =', a*b*(1000-a-b) )
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