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Does it continue? An original. (Posted on 2017-10-20) Difficulty: 3 of 5
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    
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Solution computer solution | Comment 2 of 7 |
As there were two exceptions to begin with, one would imagine there'd be more, and indeed:

 n  diff  square root
    from
    next
    square

 1   0   0 
 2   2   1.4142135623730950487 
 3   3   1.7320508075688772935 
 4   1   1.0 
 5   1   1.0 
 6   9   3.0 
 7   1   1.0 
 8   81   9.0 
 9   729   27.0 
 10   225   15.0 
 11   324   18.0 
 12   39169   197.9115964262832174277 
 13   82944   288.0 
 14   176400   420.0 
 15   215296   464.0 
 16   3444736   1856.0 
 17   26167684   5115.4358563078474007459 
 18   114349225   10693.4197055946513522096 
 19   255004929   15968.873754902065728836 
 20   1158920361   34042.9193959625031451282 
 
 

    5   open "factsq.txt" for output as #2
   10   for B=1 to 20
   20     F=!(B)
   30     Sq=(-int(-sqrt(F)))^2
   40     print B,Sq-F,sqrt(Sq-F)
   41     print #2,B,Sq-F,sqrt(Sq-F)
   50   next
   60   close #2


  Posted by Charlie on 2017-10-20 11:04:42
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