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Duodecimal Digits (Posted on 2011-09-22) Difficulty: 2 of 5
Determine the possible nonzero units digits of a duodecimal positive integer n such that:

Each of n and n+2 is a prime number, and:

n+2 is expressible as the sum of squares of two positive itegers.

See The Solution Submitted by K Sengupta    
Rating: 4.5000 (2 votes)

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Solution heuristic computer exploration -- no proof | Comment 1 of 5

The following program finds that among the 12,792 such pairs through 6104927, 6104929=112225+5992704, the final duodecimal digit of n+2 is 1 except for the case where n+2 = 5, where obviously the only digit is 5.

This means that for n, the only values are 3 and B (the latter representing base-10 11), with there being only one instance of the 3, which is 3 itself.

Note that the program's N is the puzzle's n+2:

  10       loop
  20         Prev=N
  30         N=nxtprm(N)
  40         if N-Prev=2 then
  50            :Good=0
  60            :for I=1 to int(sqrt(N)/2)
  70              :Sq1=I*I
  80              :Sq2=N-Sq1
  90              :Sr2=int(sqrt(Sq2)+0.5)
 100              :if Sr2*Sr2=Sq2 then Good=1:cancel for:goto 120:endif
 110            :next I
 120            :if Good=1 then inc Ct:print N,N @ 12,Ct,Sq1;Sq2:endif
 125            :if Good=1 and N @ 12<>1 and N>5 then stop:endif
 130         :endif
 140       endloop

  Posted by Charlie on 2011-09-22 15:21:02
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