Take any odd number and square it. It will invariably be a multiple of 8 plus 1. So (odd)^2=8n+1 where n is an integer. Show why this is always so. Also show what the pattern for n is.
(In reply to
Square of an Odd Number by Douglas Johnson)
Although your basic math is right, your definition of m! is wrong, so the equations N² = 8m! +1 and n = m! are wrong.
m! = m(m  1)(m 2)...(3)(2)(1)
m(m+1)/2 = T(m), the mth triangle number (1,3,6,10...)
If you substitute T(m) everywhere that you used m!, then your answer would be correct.
Incidently T(m) is related to m!. T(m  1) = C(m,2), where C(a,b) is the combination (usually indicated by stacking a over b like a fraction without the division bar, and enclosing them in parentases) and is equal to a!/b!(a  b)!

Posted by TomM
on 20021007 19:16:26 