How do I prove the following?
If
. . .\(\displaystyle \mbox{i) }\,a^2 \,+\, b^2\, \ne\, c^2\, +\,d^2\,\)
. . .\(\displaystyle \mbox{ii) }\,a\, \ne\, b\,\mbox{ and }\, c\, \ne\, d\)
. . .\(\displaystyle \mbox{iii) }\,a,\, b,\, c,\, d\, \ge\, 0\,\)
. . .\(\displaystyle \mbox{iv) }\,L\, \ge\, a,\, b,\, c,\, d,\)
then \(\displaystyle \,(a\,+\,L)^2 \,+\, (b\, +\,L)^2\, \ne\, (c\, +\, L)^2 \,+\, (d\, +\,L)^2\)
If \(\displaystyle (a+L)^2 , (b+L)^2, (c+L)^2, (d+L)^2\) are all greater than \(\displaystyle a^2,b^2,c^2,d^2\) what is optional L such that it can happen in general the value of L is always greater than the values taken by a,b,c,d in general. If a,b,c,d can take values from 0 to some k-1 for which value of L can we state the above inequallity is true. It is understood counter examples are possible is there any general way to find a suitable L given k and all pairs squares 0 to k-1 allowed
If
. . .\(\displaystyle \mbox{i) }\,a^2 \,+\, b^2\, \ne\, c^2\, +\,d^2\,\)
. . .\(\displaystyle \mbox{ii) }\,a\, \ne\, b\,\mbox{ and }\, c\, \ne\, d\)
. . .\(\displaystyle \mbox{iii) }\,a,\, b,\, c,\, d\, \ge\, 0\,\)
. . .\(\displaystyle \mbox{iv) }\,L\, \ge\, a,\, b,\, c,\, d,\)
then \(\displaystyle \,(a\,+\,L)^2 \,+\, (b\, +\,L)^2\, \ne\, (c\, +\, L)^2 \,+\, (d\, +\,L)^2\)
If \(\displaystyle (a+L)^2 , (b+L)^2, (c+L)^2, (d+L)^2\) are all greater than \(\displaystyle a^2,b^2,c^2,d^2\) what is optional L such that it can happen in general the value of L is always greater than the values taken by a,b,c,d in general. If a,b,c,d can take values from 0 to some k-1 for which value of L can we state the above inequallity is true. It is understood counter examples are possible is there any general way to find a suitable L given k and all pairs squares 0 to k-1 allowed
Last edited by a moderator: