Problems & Puzzles: Puzzles
Puzzle 111. Spoof odd Perfect numbers
It's almost believed that there is not any odd perfect number. But what about "almost" or "quasi" or "spoof" odd perfect numbers?
Descartes found one odd Spoof Perfect Number: 18.104.22.168.22021, that is odd & perfect only if you suppose (incorrectly) that 22021 is prime.
You can verify the above statement if you remember that:
1. Find the least and/or other odd Spoof Perfect Numbers (*)
On the contrary, as you know, there are many even Perfect Numbers, at least as many as Mersenne prime numbers (38 at the moment). But, is there any even Spoof Perfect Number? The answer is "yes", and in this case there are many of them.
The following are the least 3 examples of even Spoof Perfect Number when you suppose incorrectly that one of its factor is prime
Find the least (if any) even
Spoof Perfect Numbers such that you
suppose incorrectly that two
factors are primes
At least one contribution for this puzzle!
Polly T. Wang sent (7/11/2002) the following:
One day after he also sent this remarkable solution to question 3:
As Leech, now I can ask: is 390405312000 the least one of these even completely spoof perfect numbers?
As a matter of fact I sent an email (9/11/02) to some of the most assiduous puzzlers of these pages:
The same day I sent this email Shyam Sunder Gupta, David Terr & L. T. Pebody found one and the same smaller solution:
907200 = 4*6*8*9*15*35
Terr solved the question c):
I'm asking him to work this method step by step with the present result.
And one day later, he finally added:
So, now we know the minimal even completely spoof perfect number: 907200 a product of 6 spoof primes, and more than that... this happened thanks to the challenging result from Wang and to the smart work of Shyam and David.
I have challenged to Polly, David and Shyam to find more odd spoof perfect numbers and a systematic approach to produce them. In return to my invitation Shyam asks to look for odd & completely spoof perfect numbers or at least bounds below which such a number can not exist; but he believes that there are none.