What is the asymptotic density of odd perfect numbers?
548895
It isn't known isthere are *any* odd perfect numbers, but even if there are, the asymptotic density is zero.
129045
Even numbers are divisible by 2 (2,4,6,8,10, etc) whereas odd are not (1,3,5,7,9, etc)
579799
Independent Order of Odd Fellows
1172244
I don't know where this law of numbers came from....\n\nBut...\n\nMaybe what it should say is..."Any odd number with any given power can be expressed as a difference of one consecutive set of two perfect squares."...since that statement is true...\n\nThe difference between two consecutive squares is, in fact, the sequence of odd numbers.\n\nThis is interesting too, because all prime numbers will be included in this sequence, except for 2.
5768
I personally think they don't exist.  Here's what Wikipedia has to say, "It is unknown whether there are any odd perfect numbers. Various results have been obtained, but none that have helped to locate one or otherwise resolve the question of their existence. Recently, Carl Pomerance and Joshua Zelinsky have both presented heuristics which strongly suggest that no odd perfect numbers exist."
676633
54\n*45\n____\n\n 270\n\n2160\n_____\n\n2430
279409
Find every number between 101 and 999 that is divisible by 7 and then add them together.  Hmmm......like I really am going to do that.
103412
If you have the number of divisors N, take K to be for example p^(N-1), for an odd prime p. Then your K would have the desired number N of divisors, given in the conditions. But, you have found one such way, and cannot guarantee that it is unique, there may be several such numbers. Therefore, you cannot uniquely determine the unknown number, there will always be several possibilities for that number, but you can always find at least one as follows from Euler's multiplicative function in number theory that counts the number of divisors.\n\nF(N)=F(P_1^k1*...*P_n^Kn)=F(p_1^k1)*...F(p_n^kn).\nHope that helps.
95232
I assume that you want a project in real analysis that you can work with to get your BS in math, right?\n\nMy counsel would be to talk to your advisor, and see if there is anything that would interest you, that your advisor would suggest.\n\nFrankly, from my limited knowledge, you're going to need at least a masters level training in math to do anything really original...\n\nOr, for that matter, have any chance of thinking up something that hasn't been thought of before.\n\nHmmm....  One thought, a spinoff of the Goldbach Conjecture.\n\nIt turns out that the number of primes less than or equal to x is x/ln(x) where ln(x) is the natural or Napierian logarithm of x.  Now, based on pure empirical computations, it appears that the number of ways the even number x can be expressed as the sum of two primes looks like it may be about x/(4*ln(x)^2); I did a least squares fit to that for all even numbers less than about 250,000 about 30 years ago, and found that it did, roughly, hold.\n\nMaybe you could try proving that: as far as I know, Fletcher's Conjecture (the first fellow who tackled the question was Rod Fletcher, an old friend of mine) has never been published, or published on.  So, you might have a go at it.
927457
Is there an infinite number of twin primes? (i.e. N and N+2 are both prime).\n\nFor other problems, see:\nhttp://mathworld.wolfram.com/UnsolvedProblems.html
22351
Many have tried, yet none have found a formula. As noted on MathWorld,\n\n"Euler commented "Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the mind will never penetrate" (Havil 2003, p. 163). In a 1975 lecture, D. Zagier commented "There are two facts about the distribution of prime numbers of which I hope to convince you so overwhelmingly that they will be permanently engraved in your hearts. The first is that, despite their simple definition and role as the building blocks of the natural numbers, the prime numbers grow like weeds among the natural numbers, seeming to obey no other law than that of chance, and nobody can predict where the next one will sprout. The second fact is even more astonishing, for it states just the opposite: that the prime numbers exhibit stunning regularity, that there are laws governing their behavior, and that they obey these laws with almost military precision" (Havil 2003, p. 171)."
939960
If P is an odd prime, then half of the time we have P = 1 mod 4, and the other half of the time we have P = 3 mod 4.\n\nThis is the same as saying that half of the odd primes are of the form 4n + 1, and the other half are of the form 4n + 3.
837658
To paraphrase your question, is it possible for π to contain, say, 129 successive 3's but not 130 of them?\n\nIf the digits of π are truly random then it would contain any possible sequence, not only the above special cases. This property of "normality" is conjectured, but not known to be true.\n\nThe website in the Source allows you to find sequences in the first 200 million digits of π. Example: 777777777 occurs, but 7777777777 does not occur in the first 200 million digits of π.
48638
Perfect numbers are positive integers n such that\nn==s(n), \n\nwhere s(n) is the restricted divisor function (i.e., the sum of proper divisors of n), or equivalently\nsigma(n)==2n, \n\nwhere sigma(n) is the divisor function (i.e., the sum of divisors of n including n itself). For example, the first few perfect numbers are 6, 28, 496, 8128, ... (Sloane's A000396), since\n6=1+2+3 \n28=1+2+4+7+14 \n496=1+2+4+8+16+31+62+124+248,
348700
infinite number that wiull never stop growing
103035
yes, numbers are infinate
852068
11\n\nany perfect square will have an odd number of factors