Do positive integers $a,b,c,d$ exist with the given properties?

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Inspired by this question : Amicable pairs of numbers and their product



I ask whether positive integers $a,b,c,d$ exist with the following properties :



$(1) 0<a<b<c<d $



$(2) ad=bc$



$(3) sigma(a)=sigma(d)$



$(4) sigma(b)=sigma(c)$



where $sigma(n)$ denotes the divisor-sum-function. Upto $d=200$, there is no solution, so I conjecture that not all the conditions can be satisfied.



This would answer the question , whether distinct pairs of amicable numbers can have the same product , in a negative way.




number-theory elementary-number-theory divisor-sum




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  • 1




    I don't know the answer, but you should start by thinking about the prime factorizations of $ad$ and $bc$ and the formula for $sigma.$
    – saulspatz
    Jul 22 at 13:34










  • Usually, I do not care of downvotes, but in this case I am really curious for the reason ...
    – Peter
    Jul 26 at 20:12














up vote
0
down vote

favorite












Inspired by this question : Amicable pairs of numbers and their product



I ask whether positive integers $a,b,c,d$ exist with the following properties :



$(1) 0<a<b<c<d $



$(2) ad=bc$



$(3) sigma(a)=sigma(d)$



$(4) sigma(b)=sigma(c)$



where $sigma(n)$ denotes the divisor-sum-function. Upto $d=200$, there is no solution, so I conjecture that not all the conditions can be satisfied.



This would answer the question , whether distinct pairs of amicable numbers can have the same product , in a negative way.




number-theory elementary-number-theory divisor-sum




share|cite|improve this question

















  • 1




    I don't know the answer, but you should start by thinking about the prime factorizations of $ad$ and $bc$ and the formula for $sigma.$
    – saulspatz
    Jul 22 at 13:34










  • Usually, I do not care of downvotes, but in this case I am really curious for the reason ...
    – Peter
    Jul 26 at 20:12












up vote
0
down vote

favorite









up vote
0
down vote

favorite











Inspired by this question : Amicable pairs of numbers and their product



I ask whether positive integers $a,b,c,d$ exist with the following properties :



$(1) 0<a<b<c<d $



$(2) ad=bc$



$(3) sigma(a)=sigma(d)$



$(4) sigma(b)=sigma(c)$



where $sigma(n)$ denotes the divisor-sum-function. Upto $d=200$, there is no solution, so I conjecture that not all the conditions can be satisfied.



This would answer the question , whether distinct pairs of amicable numbers can have the same product , in a negative way.




number-theory elementary-number-theory divisor-sum




share|cite|improve this question













Inspired by this question : Amicable pairs of numbers and their product



I ask whether positive integers $a,b,c,d$ exist with the following properties :



$(1) 0<a<b<c<d $



$(2) ad=bc$



$(3) sigma(a)=sigma(d)$



$(4) sigma(b)=sigma(c)$



where $sigma(n)$ denotes the divisor-sum-function. Upto $d=200$, there is no solution, so I conjecture that not all the conditions can be satisfied.



This would answer the question , whether distinct pairs of amicable numbers can have the same product , in a negative way.





number-theory elementary-number-theory divisor-sum





share|cite|improve this question












share|cite|improve this question




share|cite|improve this question








edited Jul 22 at 13:03
























asked Jul 22 at 12:57









Peter

45k938119




45k938119







  • 1




    I don't know the answer, but you should start by thinking about the prime factorizations of $ad$ and $bc$ and the formula for $sigma.$
    – saulspatz
    Jul 22 at 13:34










  • Usually, I do not care of downvotes, but in this case I am really curious for the reason ...
    – Peter
    Jul 26 at 20:12












  • 1




    I don't know the answer, but you should start by thinking about the prime factorizations of $ad$ and $bc$ and the formula for $sigma.$
    – saulspatz
    Jul 22 at 13:34










  • Usually, I do not care of downvotes, but in this case I am really curious for the reason ...
    – Peter
    Jul 26 at 20:12







1




1




I don't know the answer, but you should start by thinking about the prime factorizations of $ad$ and $bc$ and the formula for $sigma.$
– saulspatz
Jul 22 at 13:34




I don't know the answer, but you should start by thinking about the prime factorizations of $ad$ and $bc$ and the formula for $sigma.$
– saulspatz
Jul 22 at 13:34












Usually, I do not care of downvotes, but in this case I am really curious for the reason ...
– Peter
Jul 26 at 20:12




Usually, I do not care of downvotes, but in this case I am really curious for the reason ...
– Peter
Jul 26 at 20:12










1 Answer
1






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$a=210$, $b=310$, $c=357$, $d=527$.






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  • OK, so this cannot be used for the proof.
    – Peter
    Jul 22 at 14:00










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1 Answer
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active

oldest

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1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes








up vote
0
down vote



accepted










$a=210$, $b=310$, $c=357$, $d=527$.






share|cite|improve this answer





















  • OK, so this cannot be used for the proof.
    – Peter
    Jul 22 at 14:00














up vote
0
down vote



accepted










$a=210$, $b=310$, $c=357$, $d=527$.






share|cite|improve this answer





















  • OK, so this cannot be used for the proof.
    – Peter
    Jul 22 at 14:00












up vote
0
down vote



accepted







up vote
0
down vote



accepted






$a=210$, $b=310$, $c=357$, $d=527$.






share|cite|improve this answer













$a=210$, $b=310$, $c=357$, $d=527$.







share|cite|improve this answer













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share|cite|improve this answer











answered Jul 22 at 13:36









Gerry Myerson

143k7144294




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  • OK, so this cannot be used for the proof.
    – Peter
    Jul 22 at 14:00
















  • OK, so this cannot be used for the proof.
    – Peter
    Jul 22 at 14:00















OK, so this cannot be used for the proof.
– Peter
Jul 22 at 14:00




OK, so this cannot be used for the proof.
– Peter
Jul 22 at 14:00












 

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