2!+
2=4=2
2
3!+
3=9=3
2
4!+
1=25=5
2
5!+
1=121=11
2
6!+
9=729=27
2
7!+
1=5041=71
2
8!+
81=40401=201
2
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.
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
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Posted by Charlie
on 2017-10-20 11:04:42 |
computer solution
