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 Does it continue? An original. (Posted on 2017-10-20)
2!+2=4=22
3!+3=9=32
4!+1=25=52
5!+1=121=112
6!+9=729=272
7!+1=5041=712
8!+81=40401=2012

In the above examples, the number that must be added to each factorial to reach the first perfect square beyond it is itself a perfect square. (With the exception of 2! and 3!)

Make a conjecture about whether the observed pattern continues before you try to prove it or find a counterexample.

 No Solution Yet Submitted by Jer No Rating

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 No Subject | Comment 5 of 7 |
'So now your question is : Is every factorial number a difference of two square numbers?'

For n >= 4, n! is divisible by 4.

n! = 4k = (k+1)^2 - (k-1)^2.

In fact, for big enough n there are many solutions.

Say the product of 2a and 2b = n!.  Then n! = 4ab = (a+b)^2 - (a-b)^2.

 Posted by xdog on 2017-10-20 13:46:50

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