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:
3^{2}.7^{2}.11^{2}.13^{2}.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 = p^{a}.q^{b}...,
then s(n)=[(p^{a+1}1)/(p1)].[(q^{b+1}1)/(q1)]....

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 =2^{3}*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.
4445, 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.
***
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?
***
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.
***
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.
***
