Twice the square of the first subtracted from the square of the second is -167, what are the two integers?

1 Answer
Jul 1, 2015

Answer:

Even if we assume the integers are both positive, there are an infinite number of solutions to this question. The minimal (positive) values are
#(11,12)#

Explanation:

If the first integer is #x# and the second integer is #y#

#y^2-2x^2 = -167#

#y^2 = 2x^2-167#

#y = +-sqrt(2x^2-167)#
#color(white)("XXXX")#(from here on, I will limit my answer to positive values)

if #y# is an integer
#rArr 2x^2-167 = k^2# for some integer #k#

We could limit our search by noting that #k# must be odd.

Since #x# is an integer
#color(white)("XXXX")##(k^2-167)/2# must also be an integer

Unfortunately there are lots of solutions for #k# that satisfy the stated conditions:

#{:(k,,first,second), (11,,12,11), (15,,14,15), (81,,58,81), (101,,72,101), (475,,336,475), (591,,418,591) :}#
are the values that I found for #k<1000#
and all of these satisfy the given conditions.

(...and, yes, I know #k=y#).