Posted 3 Mar 2014 (edited) All the numbers below should be assumed to be positive integers. Definition. An abundant number is an integer n whose divisors add up to more than In. Definition. A perfect number is an integer n whose divisors add up to exactly In. Definition. A deficient number is an integer n whose divisors add up to less than In. Example. 12 is an abundant number, because 1 + 2 + 3+ 4 + 6+12 = 28 and 28 > 2x12. However, 14 is a deficient number, because 1 + 2 + 7 + 14 = 24, and 24 < 2 x 14. Your task is to consider the following conjectures and determine, with proofs, whether they are true or false. Conjecture 1. A number is abundant if and only if it is a multiple of 6. Conjecture 2. If n is perfect, then kn is abundant for any k in N. Conjecture 3. If p1 and p2 are primes, then p1/p2 is abundant. Conjecture 4. If n is deficient, then every divisor of n is deficient. Conjecture 5. If n and m are abundant, then n + m is abundant. Conjecture 6. If n and m are abundant, then nm is abundant. Conjecture 7. If n is abundant, then n is not of the form pm for some natural m and prime p. Edited 3 Mar 2014 by BMAD 0 Share this post Link to post Share on other sites

0 Posted 4 Mar 2014 False, as proven by phil1882. A smaller example is 40 < 1+2+4+5+8+10+20. True for k>1. For each divisor d_{i} of n, consider the larger number kd_{i} which is a divisor of kn. Sum(kd_{i}) = k*Sum(d_{i}) = 2kn, but 1 is also a divisor of kn which has not yet been accounted for. So the sum of kn's divisors is at least 2kn+1, which makes kn abundant. Not sure what you mean. Should it be p_{1}*p_{2}? In that case, 2*3 = 6 is perfect, all others are deficient. If p_{1} = p_{2}, the divisors of p_{1}*p_{2} are 1, p_{1}, and p_{1}^{2}. The sum 1+p_{1}+p_{1}^{2} < p_{1}+p_{1}+p_{1}^{2} = 2*p_{1}+p_{1}^{2} </= p_{1}*p_{1}+p_{1}^{2} = 2*p_{1}^{2} = 2*p_{1}*p_{2}, so p_{1}*p_{2} is deficient. If p_{1} < p_{2}, the divisors of p_{1}*p_{2} are 1, p_{1}, p_{2}, and p_{1}*p_{2}. The sum 1+p_{1}+p_{2}+p_{1}*p_{2} </= p_{2}+p_{2}+p_{1}*p_{2} = 2p_{2}+p_{1}*p_{2} </= p_{1}*p_{2}+p_{1}*p_{2} = 2*p_{1}*p_{2}, with equality iff p_{1} = 2 and p_{2} = 3. True. The negation is equivalent to "there is a non-deficient positive integer n and a positive integer k such that kn is deficient", which is easily disproven using the method in my proof for conjecture 2. False. 40 and 12 are abundant, but 40+12 = 52 > 1+2+4+13+26 is deficient. True. If n is abundant, then nm is abundant for any m. False. Every n>1, abundant or otherwise, has a prime factor p, and therefore is of the form pm for some natural m. 0 Share this post Link to post Share on other sites

0 Posted 3 Mar 2014 The definitions should be in comparison of 2n not in. All the numbers below should be assumed to be positive integers. Definition. An abundant number is an integer n whose divisors add up to more than In. Definition. A perfect number is an integer n whose divisors add up to exactly In. Definition. A deficient number is an integer n whose divisors add up to less than In. Example. 12 is an abundant number, because 1 + 2 + 3+ 4 + 6+12 = 28 and 28 > 2x12. However, 14 is a deficient number, because 1 + 2 + 7 + 14 = 24, and 24 < 2 x 14. Your task is to consider the following conjectures and determine, with proofs, whether they are true or false. Conjecture 1. A number is abundant if and only if it is a multiple of 6. Conjecture 2. If n is perfect, then kn is abundant for any k in N. Conjecture 3. If p1 and p2 are primes, then p1/p2 is abundant. Conjecture 4. If n is deficient, then every divisor of n is deficient. Conjecture 5. If n and m are abundant, then n + m is abundant. Conjecture 6. If n and m are abundant, then nm is abundant. Conjecture 7. If n is abundant, then n is not of the form pm for some natural m and prime p. 0 Share this post Link to post Share on other sites

0 Posted 3 Mar 2014 If p1 and p2 are primes then p1/p2 cannot be an integer and therefore not abundant (Definition 1). 0 Share this post Link to post Share on other sites

0 Posted 4 Mar 2014 i think he ment multiplication not division there. 1) false. 28000 :: 1 +2 +4 +5 +7 +10 +14 +16 +25 +28 +32 +125 +224 875 +1000+1120+1750+2000+2800 +4000 +5600 +7000 +14000 = 40638, definitely abundant. 2) true, though i have no idea how to prove it at this point 3) true, with the exception of 2 and 3. multiplication of two primes will produce a larger result than addition. 4) true, follows from 2 assuming that can be proven. 0 Share this post Link to post Share on other sites

0 Posted 4 Mar 2014 nicely done for 2.for seven i'm not sure if thats what hes asking. i think he means that if n is abundant, then it must not have one prime raised to some power. that is if n is abundant, it must be something like 2^4*3^2*5^3.... but i agree based on 2 its definitely false. 0 Share this post Link to post Share on other sites

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