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For a certain positive integer m, the equation has 137 solutions in integers n. Find m.

May 3, 2020

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This seems difficult.......!!!

Let's see if we can find a pattern.....

floor [sqrt (1) ]  to floor [ sqrt (3)]  have the same result  = 1  ⇒ 3 occurences

floor [sqrt (4) ] to floor [ sqrt (8)] have the same result = 2  ⇒ 5 occurences

floor [ sqrt (9)] to floor [ sqrt (15)] have the same result = 3  ⇒ 7 occurences

floor  [ sqrt (16) to floor [sqrt (24)]  have the same result = 4 ⇒ 9 occurences

floor [sqrt (25)] to floor [sqrt (35)] have the same result = 5 ⇒ 11 occurences

So...it appears  that    2(result)  + 1  =  the number of occurences

And we need 137 occurences

So  we  need to solve this:

2 (result)  + 1  = 137      subtract 1 from both sides

2 (result) = 136      divide both sides by 2

result  = 68

But we need to square this to find   the first   "n"  = 68^2  = 4624

And  the last "n"  =  4624 + 136  = 4760

Check

floor [ sqrt (4623)]  = 67

floor [ sqrt (6084)]  = 68

floor [sqrt (4760)]  = 68

floor [ sqrt (4761)]  =  69

So  m   =  68   May 3, 2020