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: 32.72.112.132.22021, that is odd & perfect only if you suppose (incorrectly) that 22021 is prime.

You can verify the above statement if you remember that:

  • If n = pa.qb..., then s(n)=[(pa+1-1)/(p-1)].[(qb+1-1)/(q-1)]....

  • If n is perfect then s(n) = 2.n

Questions:

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   

60 =3*5*4 is an even Spoof Perfect Number if you suppose incorrectly that 4 is a prime
90 =2*5*9 is an even Spoof Perfect Number if you suppose incorrectly that 9 is a prime
120 =23*15 is an even Spoof Perfect Number if you suppose incorrectly that 15 is a prime
and many more... 

2. Find the least (if any) even Spoof Perfect Numbers such that you suppose incorrectly that two factors are primes
3. Find the least (if any) even Spoof Perfect Numbers such that you suppose incorrectly that all the factors are primes

________
(*)This question was first asked by John Leech according to B1, pp. 44-45, UPiNT2, R.K.Guy

Solution

At least one contribution for this puzzle!

Polly T. Wang sent (7/11/2002) the following:

840=4*5*6*7 is an even spoof perfect number if you suppose incorrectly that 4 and 6 are primes.

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One day after he also sent this remarkable solution to question 3:

390405312000 = 4*8*9*10*15*22*46*94*95 is an even spoof perfect number if you suppose incorrectly that all the factors are primes.

As Leech, now I can ask: is 390405312000 the least one of these even completely spoof perfect numbers?

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As a matter of fact I sent an email (9/11/02) to some of the most assiduous puzzlers of these pages:

I wonder if some of you... would like to try with the following questions related to the Wang's result: a) is there a smaller even completely spoof perfect number? b) is there a even completely spoof perfect numbers with less composite factors (less than 9) than the found by Wang? c) is there a regular approach in order to produce these kind of numbers?

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):

The method for finding such numbers is similar to the method for finding multiperfect numbers: Write down a small composite number (spoof prime factor) p and above it write down p+1. Simplify the resulting fraction. Choose the next spoof prime to be a small multiple of the simplified numerator. Continue until the resulting fraction is equal to 2.

I'm asking him to work this method step by step with the present result.

Shyam added:

...This is the only solution found (less than 4*10^11) which consists of 6 composite factors... I have found number of solutions(29 which are less than 4*10^11) with 7 composite factors like 172368000=4*6*8*9*14*75*95. I have found 156 solutions (which are less than 4*10^11) with 8 composite factors like 4750099200 = 4*6*8*10*33*34*35*63.

...I also state the following conjecture: "For a given number of Composite factors, the number of even completely spoof perfect numbers is finite." I feel that this is an important conjecture but proof may be difficult though may be more interesting than the conjecture itself.

And one day later, he finally added:

I have confirmed and proved also that there can not be a solution with 5 or less number of composite factors. Also the solution I gave with 6 composite factors is smallest.

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.

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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.

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