A set X is said to be countable if the function f : N --> X is onto, where N is the set of natural numbers. An odd number M is said to be perfect if M is equal to the sum of its positive proper divisors (that is, including 1 but excluding M itself). An odd perfect number, according to an age-old result by Euler, has to have the form (p^k)(m^2), where gcd(p, m) = gcd(p^k, m^2) = 1 and p is congruent to k and also congruent to 1 modulo 4. If one exists, then an odd perfect number must be at least 10^500 (www.oddperfect.org), must have at least 9 distinct prime factors (Nielsen 2006) and must be a sum of two squares (Stuyvaert 1896).
2006-06-08 22:50:50 · 9 answers · asked by JoseABDris 2 in Science & Mathematics ➔ Mathematics
They are obviously countable. Remember that the set of perfect numbers is a subset of the set of integers (and more specifically, natural numbers). Since the set of integers is a countable set, and any subset of a countable set is still countable, the set of perfect numbers must obviously be countable.
After rereading your question, I got a little better idea of what you are asking. It seems that nobody knows if the set of odd perfect numbers is infinite (or even non-null) from the website www.oddperfect.org. Therefore since I am one of these nobodies, I also don't know. But I can tell you for certain (as stated above) that the set of odd perfect numbers is countable (I am using the definition of countable in the since that the set can be finite OR infinite).
2006-06-08 23:34:18 · answer #1 · answered by Eulercrosser 4 · 0⤊ 1⤋
Every subset of the natural numbers is countable. The real question is whether the set of odd perfect numbers is non-empty. Your definition of countability is a bit off since you make a statement about 'the' function, when there can be many. All that is needed is *one* onto function from the set of naturals for a set to be countable.
2006-06-09 00:33:58 · answer #2 · answered by mathematician 7 · 0⤊ 0⤋
Since the set of whole numbers in totally uncountable, logic dictates that any subset of the whole numbers ( as in having a beginning but apparently no end) is also uncountable simply because all numbers have yet to be discovered.
I'm no math expert but to me, this is common sense
2006-06-08 22:56:13 · answer #3 · answered by Celestial Dragon 3 · 0⤊ 0⤋
India had a near perfect match after Australia's first innings batting,i thought it was good first innings from Aussies but India scored so fast thanks to Dhawan and co and Australia's second innings capitulation that happened so quickly on the last day cannot be explained even by Ian Chappell.Batting for India clicked so well in this series,they had a near perfect match.Australia should've drawn this match but they ended up losing on the last day thanks to some poor batting in second innings.
2016-03-26 23:05:14 · answer #4 · answered by ? 4 · 0⤊ 0⤋
I don't know. I guess not because odd numbers are infinite. You may count upto a limit but then if you keep trying one more will pop up.
2006-06-20 01:21:52 · answer #5 · answered by nayanmange 4 · 0⤊ 0⤋
any given number having any given power can be expressed as difference of at least one set of two perfect squares. if the power is one then any number in this universe can be expressed as difference of at least one set of two perfect squares.
2006-06-09 01:41:07 · answer #6 · answered by rajesh bhowmick 2 · 0⤊ 0⤋
Yes it is
2006-06-08 22:57:36 · answer #7 · answered by ag_iitkgp 7 · 0⤊ 0⤋
00000111
2006-06-22 13:24:37 · answer #8 · answered by Mike 3 · 0⤊ 1⤋
i dont know i guess
2006-06-08 22:54:42 · answer #9 · answered by tarenirator 2 · 0⤊ 1⤋