Highest power of given primen number in all binomial coefficients of n [closed]

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Given prime number p and integer n, for what value of i where 0$leq$i$leq$n, the highest of p in $n choose i$ occurs.






prime-numbers binomial-coefficients combinations







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closed as off-topic by Nosrati, Leucippus, max_zorn, José Carlos Santos, amWhy Aug 31 at 9:41


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    Given prime number p and integer n, for what value of i where 0$leq$i$leq$n, the highest of p in $n choose i$ occurs.






    prime-numbers binomial-coefficients combinations







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    closed as off-topic by Nosrati, Leucippus, max_zorn, José Carlos Santos, amWhy Aug 31 at 9:41


    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." – Nosrati, Leucippus, max_zorn, José Carlos Santos, amWhy
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      Given prime number p and integer n, for what value of i where 0$leq$i$leq$n, the highest of p in $n choose i$ occurs.






      prime-numbers binomial-coefficients combinations







      share|cite|improve this question













      Given prime number p and integer n, for what value of i where 0$leq$i$leq$n, the highest of p in $n choose i$ occurs.






      prime-numbers binomial-coefficients combinations




      prime-numbers binomial-coefficients combinations






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      asked Aug 31 at 5:08









      Sai Satwik Kuppili

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      closed as off-topic by Nosrati, Leucippus, max_zorn, José Carlos Santos, amWhy Aug 31 at 9:41


      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." – Nosrati, Leucippus, max_zorn, José Carlos Santos, amWhy
      If this question can be reworded to fit the rules in the help center, please edit the question.




      closed as off-topic by Nosrati, Leucippus, max_zorn, José Carlos Santos, amWhy Aug 31 at 9:41


      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." – Nosrati, Leucippus, max_zorn, José Carlos Santos, amWhy
      If this question can be reworded to fit the rules in the help center, please edit the question.




















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          Kummer's theorem on binomial coefficients tells you that the exponent of $p$ occurring in $n choose i$ is the number of carries when adding $n-i$ to $i$ in base $p$. So you want to choose $i$ to maximize this number of carries. The maximum number of carries will occur with $i = p^m-1$
          where $p^m le n < p^m+1$ (this generally won't be the only solution).






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            1 Answer
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            active

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            1 Answer
            1






            active

            oldest

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            active

            oldest

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            active

            oldest

            votes








            up vote
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            down vote



            accepted










            Kummer's theorem on binomial coefficients tells you that the exponent of $p$ occurring in $n choose i$ is the number of carries when adding $n-i$ to $i$ in base $p$. So you want to choose $i$ to maximize this number of carries. The maximum number of carries will occur with $i = p^m-1$
            where $p^m le n < p^m+1$ (this generally won't be the only solution).






            share|cite|improve this answer
























              up vote
              0
              down vote



              accepted










              Kummer's theorem on binomial coefficients tells you that the exponent of $p$ occurring in $n choose i$ is the number of carries when adding $n-i$ to $i$ in base $p$. So you want to choose $i$ to maximize this number of carries. The maximum number of carries will occur with $i = p^m-1$
              where $p^m le n < p^m+1$ (this generally won't be the only solution).






              share|cite|improve this answer






















                up vote
                0
                down vote



                accepted







                up vote
                0
                down vote



                accepted






                Kummer's theorem on binomial coefficients tells you that the exponent of $p$ occurring in $n choose i$ is the number of carries when adding $n-i$ to $i$ in base $p$. So you want to choose $i$ to maximize this number of carries. The maximum number of carries will occur with $i = p^m-1$
                where $p^m le n < p^m+1$ (this generally won't be the only solution).






                share|cite|improve this answer












                Kummer's theorem on binomial coefficients tells you that the exponent of $p$ occurring in $n choose i$ is the number of carries when adding $n-i$ to $i$ in base $p$. So you want to choose $i$ to maximize this number of carries. The maximum number of carries will occur with $i = p^m-1$
                where $p^m le n < p^m+1$ (this generally won't be the only solution).







                share|cite|improve this answer












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                share|cite|improve this answer










                answered Aug 31 at 5:37









                Robert Israel

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                307k22201443












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