Can anyone help with a maths question relating to Pythagorean triads. ?

Question

Prove that a sufficient condition for a positive integer to be the difference between the squares of consecutive integers is that it is odd.

Answer 1

Hi

Why can't you just answer it and say "by inspection".

Say you have 3^2 = 9, and 2^2 = 4. You then do 9-4 =5 which is odd. You take another pair like 1^2 and 2^2. You get 4-1=3 which is odd. So by inspection you can prove it or do oyu need to have it in algebra form?

Answer 2

3, 4, 5

Answer 3

I agree with 3,4,5 being pythagorean triplets but the wording of your question is a bit odd. The way you wrote it insinuates that if a number is odd then it can be written as the difference of squares of two consecutive integers. Clearly coming up with one example does not show that this condition holds true always.

If i am just reading/understanding this wrong I apologize.  

Answer 4

the pattern 3,4,5 then 5,12,13  then 7,24,25  and 9,40,41 and so on means that every odd number is part of a pythagorean triad.  But this is not a proof.  Nevertheless it shows that every odd number is part of the P triad.  Try using algebra with x then x+1 and squaring and subtracting then taking the square root to get the third number of the triad. i think it is the square root of 2x + 1 which must be every odd number (you could write them out e.g. 3 is sq rt of 2 (4) + 1, 5 is sq rt of 2 (12) +1, 7 is sq rt of 2 (24) +1

alternatively an advanced method would be to use mathematical induction