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number theory: 7 conjectures


Best Answer Rainman, 04 March 2014 - 02:27 PM

Spoiler for
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#1 BMAD

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Posted 03 March 2014 - 06:12 PM

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, 03 March 2014 - 06:13 PM.

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#2 BMAD

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Posted 03 March 2014 - 07:50 PM

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.


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#3 fabpig

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Posted 03 March 2014 - 10:30 PM

Spoiler for Conjecture 3


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#4 phil1882

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Posted 04 March 2014 - 02:51 AM

i think he ment multiplication not division there.

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#5 Rainman

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Posted 04 March 2014 - 02:27 PM   Best Answer

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#6 phil1882

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Posted 04 March 2014 - 02:37 PM

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.


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