New Number Theory Conjecture [on hold]

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I have accidentally created a conjecture in number theory. In an attempt to disprove the existence of odd perfect numbers I noticed a pattern and formulated an equation. If this equation I created can be proven to be mathematically sound, then I will have successfully disproved the existence of odd perfect numbers.

My conjecture states: N = P_N[σ1(N)/d(N)]

Where,

• N is a perfect number (generated by P)


• P_N is the Mersenne Prime exponent (2^(P_N-1)(2^(P_N) - 1))


• σ1(N) is the summation of the divisors of N (including 1 and N)


• d(N) is the number of divisors of N (including 1 and N)

For example:

N = 28

P₂₈ = 3 | 2^(3-1)(2^(3) - 1) = 28

σ1(28) = 1 + 2 + 4 + 7 + 14 + 28 = 56

d(28) = 6

28 = 3[56/6] or 28 = 28

I have tested my equation for every known perfect number and it has always been correct. My only problem now is that I am clueless on how to prove that my conjecture works for all perfect numbers. Is there any way to prove this relationship?

number-theory prime-numbers divisor-sum perfect-numbers mersenne-numbers

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edited 5 hours ago

asked 6 hours ago

Maxwell Schaffer

62

put on hold as off-topic by Elliot G, Morgan Rodgers, gammatester, Hans Lundmark, David G. Stork 6 hours ago

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Elliot G, Morgan Rodgers, gammatester, Hans Lundmark, David G. Stork

If this question can be reworded to fit the rules in the help center, please edit the question.

• 
  
  
  

  
  

  
  
  
  
  
  

  
  This could be interesting but I can’t follow it. Can you show a couple examples of all these terms and the result?
  – Randall
  6 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  For the known perfect number $N=28$, $fracσ1(N)d(N)=frac1+2+4+7+14+286=9frac13$. Does this fit into your pattern?
  – James Arathoon
  6 hours ago
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  For perfect numbers $sigma_1(N)=2N$, so the equation is really $d(N)=2P_N$, this should be an easy exercise. Of course the question should clear up that you consider only even perfect numbers here.
  – Arnaud Mortier
  3 hours ago
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  (continued) since you talk about Mersenne primes, which are known to be in $1$-$1$ correspondence with perfect numbers.
  – Arnaud Mortier
  3 hours ago
  
  

  
  
  
  

add a comment | 

up vote
1
down vote

favorite

4

I have accidentally created a conjecture in number theory. In an attempt to disprove the existence of odd perfect numbers I noticed a pattern and formulated an equation. If this equation I created can be proven to be mathematically sound, then I will have successfully disproved the existence of odd perfect numbers.

My conjecture states: N = P_N[σ1(N)/d(N)]

Where,

• N is a perfect number (generated by P)


• P_N is the Mersenne Prime exponent (2^(P_N-1)(2^(P_N) - 1))


• σ1(N) is the summation of the divisors of N (including 1 and N)


• d(N) is the number of divisors of N (including 1 and N)

For example:

N = 28

P₂₈ = 3 | 2^(3-1)(2^(3) - 1) = 28

σ1(28) = 1 + 2 + 4 + 7 + 14 + 28 = 56

d(28) = 6

28 = 3[56/6] or 28 = 28

I have tested my equation for every known perfect number and it has always been correct. My only problem now is that I am clueless on how to prove that my conjecture works for all perfect numbers. Is there any way to prove this relationship?

number-theory prime-numbers divisor-sum perfect-numbers mersenne-numbers

share|cite|improve this question

edited 5 hours ago

asked 6 hours ago

Maxwell Schaffer

62

put on hold as off-topic by Elliot G, Morgan Rodgers, gammatester, Hans Lundmark, David G. Stork 6 hours ago

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Elliot G, Morgan Rodgers, gammatester, Hans Lundmark, David G. Stork

If this question can be reworded to fit the rules in the help center, please edit the question.

• 
  
  
  

  
  

  
  
  
  
  
  

  
  This could be interesting but I can’t follow it. Can you show a couple examples of all these terms and the result?
  – Randall
  6 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  For the known perfect number $N=28$, $fracσ1(N)d(N)=frac1+2+4+7+14+286=9frac13$. Does this fit into your pattern?
  – James Arathoon
  6 hours ago
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  For perfect numbers $sigma_1(N)=2N$, so the equation is really $d(N)=2P_N$, this should be an easy exercise. Of course the question should clear up that you consider only even perfect numbers here.
  – Arnaud Mortier
  3 hours ago
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  (continued) since you talk about Mersenne primes, which are known to be in $1$-$1$ correspondence with perfect numbers.
  – Arnaud Mortier
  3 hours ago
  
  

  
  
  
  

add a comment | 

up vote
1
down vote

favorite

4

up vote
1
down vote

favorite

4

4

I have accidentally created a conjecture in number theory. In an attempt to disprove the existence of odd perfect numbers I noticed a pattern and formulated an equation. If this equation I created can be proven to be mathematically sound, then I will have successfully disproved the existence of odd perfect numbers.

My conjecture states: N = P_N[σ1(N)/d(N)]

Where,

• N is a perfect number (generated by P)


• P_N is the Mersenne Prime exponent (2^(P_N-1)(2^(P_N) - 1))


• σ1(N) is the summation of the divisors of N (including 1 and N)


• d(N) is the number of divisors of N (including 1 and N)

For example:

N = 28

P₂₈ = 3 | 2^(3-1)(2^(3) - 1) = 28

σ1(28) = 1 + 2 + 4 + 7 + 14 + 28 = 56

d(28) = 6

28 = 3[56/6] or 28 = 28

I have tested my equation for every known perfect number and it has always been correct. My only problem now is that I am clueless on how to prove that my conjecture works for all perfect numbers. Is there any way to prove this relationship?

number-theory prime-numbers divisor-sum perfect-numbers mersenne-numbers

share|cite|improve this question

edited 5 hours ago

asked 6 hours ago

Maxwell Schaffer

62

I have accidentally created a conjecture in number theory. In an attempt to disprove the existence of odd perfect numbers I noticed a pattern and formulated an equation. If this equation I created can be proven to be mathematically sound, then I will have successfully disproved the existence of odd perfect numbers.

My conjecture states: N = P_N[σ1(N)/d(N)]

Where,

• N is a perfect number (generated by P)


• P_N is the Mersenne Prime exponent (2^(P_N-1)(2^(P_N) - 1))


• σ1(N) is the summation of the divisors of N (including 1 and N)


• d(N) is the number of divisors of N (including 1 and N)

For example:

N = 28

P₂₈ = 3 | 2^(3-1)(2^(3) - 1) = 28

σ1(28) = 1 + 2 + 4 + 7 + 14 + 28 = 56

d(28) = 6

28 = 3[56/6] or 28 = 28

I have tested my equation for every known perfect number and it has always been correct. My only problem now is that I am clueless on how to prove that my conjecture works for all perfect numbers. Is there any way to prove this relationship?

number-theory prime-numbers divisor-sum perfect-numbers mersenne-numbers

share|cite|improve this question

edited 5 hours ago

asked 6 hours ago

Maxwell Schaffer

62

share|cite|improve this question

share|cite|improve this question

edited 5 hours ago

edited 5 hours ago

edited 5 hours ago

asked 6 hours ago

Maxwell Schaffer

62

asked 6 hours ago

Maxwell Schaffer

62

asked 6 hours ago

Maxwell Schaffer

62

62

put on hold as off-topic by Elliot G, Morgan Rodgers, gammatester, Hans Lundmark, David G. Stork 6 hours ago

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Elliot G, Morgan Rodgers, gammatester, Hans Lundmark, David G. Stork

If this question can be reworded to fit the rules in the help center, please edit the question.

put on hold as off-topic by Elliot G, Morgan Rodgers, gammatester, Hans Lundmark, David G. Stork 6 hours ago

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Elliot G, Morgan Rodgers, gammatester, Hans Lundmark, David G. Stork

If this question can be reworded to fit the rules in the help center, please edit the question.

• 
  
  
  

  
  

  
  
  
  
  
  

  
  This could be interesting but I can’t follow it. Can you show a couple examples of all these terms and the result?
  – Randall
  6 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  For the known perfect number $N=28$, $fracσ1(N)d(N)=frac1+2+4+7+14+286=9frac13$. Does this fit into your pattern?
  – James Arathoon
  6 hours ago
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  For perfect numbers $sigma_1(N)=2N$, so the equation is really $d(N)=2P_N$, this should be an easy exercise. Of course the question should clear up that you consider only even perfect numbers here.
  – Arnaud Mortier
  3 hours ago
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  (continued) since you talk about Mersenne primes, which are known to be in $1$-$1$ correspondence with perfect numbers.
  – Arnaud Mortier
  3 hours ago
  
  

  
  
  
  

add a comment | 

• 
  
  
  

  
  

  
  
  
  
  
  

  
  This could be interesting but I can’t follow it. Can you show a couple examples of all these terms and the result?
  – Randall
  6 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  For the known perfect number $N=28$, $fracσ1(N)d(N)=frac1+2+4+7+14+286=9frac13$. Does this fit into your pattern?
  – James Arathoon
  6 hours ago
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  For perfect numbers $sigma_1(N)=2N$, so the equation is really $d(N)=2P_N$, this should be an easy exercise. Of course the question should clear up that you consider only even perfect numbers here.
  – Arnaud Mortier
  3 hours ago
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  (continued) since you talk about Mersenne primes, which are known to be in $1$-$1$ correspondence with perfect numbers.
  – Arnaud Mortier
  3 hours ago
  
  

  
  
  
  

This could be interesting but I can’t follow it. Can you show a couple examples of all these terms and the result?
– Randall
6 hours ago

This could be interesting but I can’t follow it. Can you show a couple examples of all these terms and the result?
– Randall
6 hours ago

For the known perfect number $N=28$, $fracσ1(N)d(N)=frac1+2+4+7+14+286=9frac13$. Does this fit into your pattern?
– James Arathoon
6 hours ago

For the known perfect number $N=28$, $fracσ1(N)d(N)=frac1+2+4+7+14+286=9frac13$. Does this fit into your pattern?
– James Arathoon
6 hours ago

For perfect numbers $sigma_1(N)=2N$, so the equation is really $d(N)=2P_N$, this should be an easy exercise. Of course the question should clear up that you consider only even perfect numbers here.
– Arnaud Mortier
3 hours ago

For perfect numbers $sigma_1(N)=2N$, so the equation is really $d(N)=2P_N$, this should be an easy exercise. Of course the question should clear up that you consider only even perfect numbers here.
– Arnaud Mortier
3 hours ago

(continued) since you talk about Mersenne primes, which are known to be in $1$-$1$ correspondence with perfect numbers.
– Arnaud Mortier
3 hours ago

(continued) since you talk about Mersenne primes, which are known to be in $1$-$1$ correspondence with perfect numbers.
– Arnaud Mortier
3 hours ago

add a comment | 

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