Jump to content
BrainDen.com - Brain Teasers
  • 0

number theory: 7 conjectures


BMAD
 Share

Question

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 by BMAD
  • Upvote 1
  • Downvote 1
Link to comment
Share on other sites

5 answers to this question

Recommended Posts

  • 0

  1. False, as proven by phil1882. A smaller example is 40 < 1+2+4+5+8+10+20.
  2. True for k>1. For each divisor di of n, consider the larger number kdi which is a divisor of kn. Sum(kdi) = k*Sum(di) = 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.
  3. Not sure what you mean. Should it be p1*p2? In that case, 2*3 = 6 is perfect, all others are deficient. If p1 = p2, the divisors of p1*p2 are 1, p1, and p12. The sum 1+p1+p12 < p1+p1+p12 = 2*p1+p12 </= p1*p1+p12 = 2*p12 = 2*p1*p2, so p1*p2 is deficient. If p1 < p2, the divisors of p1*p2 are 1, p1, p2, and p1*p2. The sum 1+p1+p2+p1*p2 </= p2+p2+p1*p2 = 2p2+p1*p2 </= p1*p2+p1*p2 = 2*p1*p2, with equality iff p1 = 2 and p2 = 3.
  4. 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.
  5. False. 40 and 12 are abundant, but 40+12 = 52 > 1+2+4+13+26 is deficient.
  6. True. If n is abundant, then nm is abundant for any m.
  7. False. Every n>1, abundant or otherwise, has a prime factor p, and therefore is of the form pm for some natural m.

Link to comment
Share on other sites

  • 0

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.

Link to comment
Share on other sites

  • 0

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.

Link to comment
Share on other sites

Join the conversation

You can post now and register later. If you have an account, sign in now to post with your account.

Guest
Answer this question...

×   Pasted as rich text.   Paste as plain text instead

  Only 75 emoji are allowed.

×   Your link has been automatically embedded.   Display as a link instead

×   Your previous content has been restored.   Clear editor

×   You cannot paste images directly. Upload or insert images from URL.

Loading...
 Share

  • Recently Browsing   0 members

    • No registered users viewing this page.
×
×
  • Create New...