anonymous
  • anonymous
I need help with a proof. Prove the equation a^2=4*b+3 has no integer solutions.
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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katieb
  • katieb
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anonymous
  • anonymous
I think I'm supposed to use a proof by cases, but I don't know how to do that.
zpupster
  • zpupster
are you proving it has no integer solutions or what are the integer solutions
amistre64
  • amistre64
4b is an even number and + 3 makes it odd; but there are odd perfect squares so thats not a god idea is it prolly thinking in terms of 3 mod 4

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anonymous
  • anonymous
it should be no integer solutions
anonymous
  • anonymous
hmm...I was afraid it had to do with mod's. It's been years since I've dealt with that.
amistre64
  • amistre64
what is sqrt(4b+3) ? that might lead someplace
amistre64
  • amistre64
let a be an even integer, since integers are closed under multiplication .... a^2 is an integer (2n)^2 = 4n^2 4n^2 = 4b + 3 4n^2 - 3 = 4b n^2 - 3/4 = b thats bad for the evens at least
amistre64
  • amistre64
try it with the odds: let a = (2n+1)
anonymous
  • anonymous
ok, I think I see what you are saying. Let me try that.
anonymous
  • anonymous
so with the odds, b=n^2+n+1/4 but what does that prove?
amistre64
  • amistre64
is n^2 an integer? is n an integer? can we add or subtract a fraction and still be left with an integer?
amistre64
  • amistre64
i got b = n^2 +n - 1/2 but thats just the details :)
amistre64
  • amistre64
(2n+1)^2 = 4b + 3 4n^2 + 4n +1 = 4b + 3 4n^2 + 4n -2 = 4b n^2 + n -1/2 = b
anonymous
  • anonymous
ooops, I'm a little rusty on my basic algebra. lol
anonymous
  • anonymous
Ok, that makes sense.
amistre64
  • amistre64
lets take the fraction and make them decimals: (any integer) - .5 is not an integer (any integer) - .75 is not an integer therefore for any solution, b is not an integer
anonymous
  • anonymous
might i make a suggestion? the cases are as @amistre64 said \((4n)^2,(4n+1)^2, (4n+2)^2, (4n+3)^2\) then after you multiply note that each of these has a remainder of 0 or 1 when divided b 4, none have a remainder of 3 that is what you are trying to prove
anonymous
  • anonymous
Awesome! Thanks guys. This helps immensely
amistre64
  • amistre64
good luck :)

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