Replace the numerals 1 through 8 with ever increasing prime numbers, always using the next lowest possible that is available¹ to fulfill the criteria on the left. Then do the same for the right.

Present a series for the left, and one for the right.

7 + 8 = Square 6 = Prime 4 + 5 = Square 1 + 2 + 3 = Square

7 + 8 = Cube 6 = Prime 4 + 5 = Cube 1 + 2 + 3 = Cube

If it was required that the Right set required "1" to be the next Prime following on after that used for the "8" in the Left set, what might the Right set read, if indeed it is possible?

^1. Note, "always using the next lowest possible that is available" means that if it is next on the list it cannot be dismissed unless it is the last of a group of two or three and will not fulfill the criterion. Only then may you advance to the next.

(In reply to re: continuing on with the by Justin)

With consideration to Charlie's data and bringing those "non-events" together as Justin has indicated in his commentary, I decided to look a little further:
  P1      P2     Sum    Sqrt    sqrt/2
  17      19      36      6       3
  47      53     100     10       5
  71      73     144     12       6
 283     293     576     24      12
 881     883    1764     42      21
1151    1153    2304     48      24
1913    1931    3844     62      31

I think that while Justin has offered a worthy  explanation of this "phenomenon" I wonder if it would fully satiate Charlie, I mean, Charlie seemed to be wondering if there might be some means of predicting successive occurrences rather than having them be computer generated.

It would seem difficult to come up with some rationale as some values of P1 and P2 are prime pairs while others are more distant.

Edited on January 7, 2010, 5:32 am

Posted by brianjn on 2010-01-07 04:01:13