Can a formula that is tailor-made to incorporate answers be correct?












5














Sample problem:




Find an equation $theta(n)$ for which $theta(n)=left{ begin{array} &0, text{when } nin text{Composed} \ n, text{when } nin text{Prime} end{array} right.$




This problem is from the International Youth Math Challenge $2018$ and since they do not return marked sheets, I am unsure if my solution was correct.



My final answer was: $$theta (n)=n-ncdot text{sgn} left(prod_{i=1}^{infty} |n-p_i|right)$$ where $p_i$ is the $i^{text{th}}$ prime number. This is all I could come up with and to be honest, I am not too happy with it, because I feel like I have basically chosen something that will only give the answer I want. Is this solution correct, mathematically? Is there a better solution?



Note: $text {sgn}(n)$ is the $text{sign}$ or $text{signum}$ function and $$text{sgn}(n)=left{ begin{array} &-1; nlt 0\ 0; n=0\ 1; ngt 0 end{array} right.$$






prime-numbers contest-math







share|cite|improve this question





























    5














    Sample problem:




    Find an equation $theta(n)$ for which $theta(n)=left{ begin{array} &0, text{when } nin text{Composed} \ n, text{when } nin text{Prime} end{array} right.$




    This problem is from the International Youth Math Challenge $2018$ and since they do not return marked sheets, I am unsure if my solution was correct.



    My final answer was: $$theta (n)=n-ncdot text{sgn} left(prod_{i=1}^{infty} |n-p_i|right)$$ where $p_i$ is the $i^{text{th}}$ prime number. This is all I could come up with and to be honest, I am not too happy with it, because I feel like I have basically chosen something that will only give the answer I want. Is this solution correct, mathematically? Is there a better solution?



    Note: $text {sgn}(n)$ is the $text{sign}$ or $text{signum}$ function and $$text{sgn}(n)=left{ begin{array} &-1; nlt 0\ 0; n=0\ 1; ngt 0 end{array} right.$$






    prime-numbers contest-math







    share|cite|improve this question



























      5












      5








      5


      2





      Sample problem:




      Find an equation $theta(n)$ for which $theta(n)=left{ begin{array} &0, text{when } nin text{Composed} \ n, text{when } nin text{Prime} end{array} right.$




      This problem is from the International Youth Math Challenge $2018$ and since they do not return marked sheets, I am unsure if my solution was correct.



      My final answer was: $$theta (n)=n-ncdot text{sgn} left(prod_{i=1}^{infty} |n-p_i|right)$$ where $p_i$ is the $i^{text{th}}$ prime number. This is all I could come up with and to be honest, I am not too happy with it, because I feel like I have basically chosen something that will only give the answer I want. Is this solution correct, mathematically? Is there a better solution?



      Note: $text {sgn}(n)$ is the $text{sign}$ or $text{signum}$ function and $$text{sgn}(n)=left{ begin{array} &-1; nlt 0\ 0; n=0\ 1; ngt 0 end{array} right.$$






      prime-numbers contest-math







      share|cite|improve this question















      Sample problem:




      Find an equation $theta(n)$ for which $theta(n)=left{ begin{array} &0, text{when } nin text{Composed} \ n, text{when } nin text{Prime} end{array} right.$




      This problem is from the International Youth Math Challenge $2018$ and since they do not return marked sheets, I am unsure if my solution was correct.



      My final answer was: $$theta (n)=n-ncdot text{sgn} left(prod_{i=1}^{infty} |n-p_i|right)$$ where $p_i$ is the $i^{text{th}}$ prime number. This is all I could come up with and to be honest, I am not too happy with it, because I feel like I have basically chosen something that will only give the answer I want. Is this solution correct, mathematically? Is there a better solution?



      Note: $text {sgn}(n)$ is the $text{sign}$ or $text{signum}$ function and $$text{sgn}(n)=left{ begin{array} &-1; nlt 0\ 0; n=0\ 1; ngt 0 end{array} right.$$






      prime-numbers contest-math




      prime-numbers contest-math






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited 1 hour ago

























      asked 1 hour ago









      Mohammad Zuhair Khan

      1,4232525




      1,4232525






















          3 Answers
          3






          active

          oldest

          votes


















          2














          I'm not sure what kind of functions are allowed but here is a similar one (might be equivalent after some small changes), the differences being it's finite and doesn't require ability to select primes:



          For any positive integer $p$, define this function
          $$
          f(n,p):= leftlceil frac{n-plfloor frac{n}{p}rfloor}{n} rightrceil
          $$
          If $p$ divides $n$ then $f(n,p)=0$, otherwise $n-plfloor n/prfloorneq 0$ so $f(n,p)=1$.



          You can then use this to make the following:
          $$
          theta(n):= n - nprod_{p=2}^{n-1}f(n,p)
          $$
          If $n$ is composite then one of the $p$'s will make the product $0$ and hence $theta(n)=n$. Otherwise $n$ is prime and the product is $1$, giving $theta(n)=0$.






          share|cite|improve this answer





























            1














            Good try; but there's something that needs fixing. When $n$ is composite, the product diverges to $infty$; so you should define $text {sgn} (infty) = 1$.






            share|cite|improve this answer





















            • Thanks for the advice. But, unless I am mistaken, $+infty gt 0?$
              – Mohammad Zuhair Khan
              1 hour ago






            • 1




              @MohammadZuhairKhan Yes; just personally, I would mention that case, since often people assume that youre working in $mathbb{R}$ instead of the extended number system. I would just put it to make sure there's no chance of getting points off.
              – Ovi
              1 hour ago










            • Oh okay thank you. Ofcourse, as it is done already, I can not (and would not) update my answer. But is there any other solution for this problem? I have a suspicion that this might be an open problem, or atleast a sensible version of it will.
              – Mohammad Zuhair Khan
              59 mins ago










            • @MohammadZuhairKhan I am thinking about it
              – Ovi
              57 mins ago



















            1














            Instead of the sign function, you could potentially use the Kronecker delta, which is defined as



            $$delta_{mn}=begin{cases}
            1 & text{if }n=m\
            0 & text{if }nneq m
            end{cases}$$



            It basically compares two numbers and gives $1$ if there is a match and $0$ otherwise. By summing over such Kronecker delta's, you could build:



            $$theta(n)=nsum_{i=1}^{infty}delta_{np_i}$$



            (where $p_i$ is the $i$-th prime number). However, I'm wondering if this would be accepted because it kind of bypasses the question of "checking if $n$ is prime". Both our formulas are just a nice rewording of the "text-form" formula given in the question, so I'm not certain this was the kind of answer that was expected.






            share|cite|improve this answer





















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              3 Answers
              3






              active

              oldest

              votes








              3 Answers
              3






              active

              oldest

              votes









              active

              oldest

              votes






              active

              oldest

              votes









              2














              I'm not sure what kind of functions are allowed but here is a similar one (might be equivalent after some small changes), the differences being it's finite and doesn't require ability to select primes:



              For any positive integer $p$, define this function
              $$
              f(n,p):= leftlceil frac{n-plfloor frac{n}{p}rfloor}{n} rightrceil
              $$
              If $p$ divides $n$ then $f(n,p)=0$, otherwise $n-plfloor n/prfloorneq 0$ so $f(n,p)=1$.



              You can then use this to make the following:
              $$
              theta(n):= n - nprod_{p=2}^{n-1}f(n,p)
              $$
              If $n$ is composite then one of the $p$'s will make the product $0$ and hence $theta(n)=n$. Otherwise $n$ is prime and the product is $1$, giving $theta(n)=0$.






              share|cite|improve this answer


























                2














                I'm not sure what kind of functions are allowed but here is a similar one (might be equivalent after some small changes), the differences being it's finite and doesn't require ability to select primes:



                For any positive integer $p$, define this function
                $$
                f(n,p):= leftlceil frac{n-plfloor frac{n}{p}rfloor}{n} rightrceil
                $$
                If $p$ divides $n$ then $f(n,p)=0$, otherwise $n-plfloor n/prfloorneq 0$ so $f(n,p)=1$.



                You can then use this to make the following:
                $$
                theta(n):= n - nprod_{p=2}^{n-1}f(n,p)
                $$
                If $n$ is composite then one of the $p$'s will make the product $0$ and hence $theta(n)=n$. Otherwise $n$ is prime and the product is $1$, giving $theta(n)=0$.






                share|cite|improve this answer
























                  2












                  2








                  2






                  I'm not sure what kind of functions are allowed but here is a similar one (might be equivalent after some small changes), the differences being it's finite and doesn't require ability to select primes:



                  For any positive integer $p$, define this function
                  $$
                  f(n,p):= leftlceil frac{n-plfloor frac{n}{p}rfloor}{n} rightrceil
                  $$
                  If $p$ divides $n$ then $f(n,p)=0$, otherwise $n-plfloor n/prfloorneq 0$ so $f(n,p)=1$.



                  You can then use this to make the following:
                  $$
                  theta(n):= n - nprod_{p=2}^{n-1}f(n,p)
                  $$
                  If $n$ is composite then one of the $p$'s will make the product $0$ and hence $theta(n)=n$. Otherwise $n$ is prime and the product is $1$, giving $theta(n)=0$.






                  share|cite|improve this answer












                  I'm not sure what kind of functions are allowed but here is a similar one (might be equivalent after some small changes), the differences being it's finite and doesn't require ability to select primes:



                  For any positive integer $p$, define this function
                  $$
                  f(n,p):= leftlceil frac{n-plfloor frac{n}{p}rfloor}{n} rightrceil
                  $$
                  If $p$ divides $n$ then $f(n,p)=0$, otherwise $n-plfloor n/prfloorneq 0$ so $f(n,p)=1$.



                  You can then use this to make the following:
                  $$
                  theta(n):= n - nprod_{p=2}^{n-1}f(n,p)
                  $$
                  If $n$ is composite then one of the $p$'s will make the product $0$ and hence $theta(n)=n$. Otherwise $n$ is prime and the product is $1$, giving $theta(n)=0$.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered 21 mins ago









                  Yong Hao Ng

                  3,1891220




                  3,1891220























                      1














                      Good try; but there's something that needs fixing. When $n$ is composite, the product diverges to $infty$; so you should define $text {sgn} (infty) = 1$.






                      share|cite|improve this answer





















                      • Thanks for the advice. But, unless I am mistaken, $+infty gt 0?$
                        – Mohammad Zuhair Khan
                        1 hour ago






                      • 1




                        @MohammadZuhairKhan Yes; just personally, I would mention that case, since often people assume that youre working in $mathbb{R}$ instead of the extended number system. I would just put it to make sure there's no chance of getting points off.
                        – Ovi
                        1 hour ago










                      • Oh okay thank you. Ofcourse, as it is done already, I can not (and would not) update my answer. But is there any other solution for this problem? I have a suspicion that this might be an open problem, or atleast a sensible version of it will.
                        – Mohammad Zuhair Khan
                        59 mins ago










                      • @MohammadZuhairKhan I am thinking about it
                        – Ovi
                        57 mins ago
















                      1














                      Good try; but there's something that needs fixing. When $n$ is composite, the product diverges to $infty$; so you should define $text {sgn} (infty) = 1$.






                      share|cite|improve this answer





















                      • Thanks for the advice. But, unless I am mistaken, $+infty gt 0?$
                        – Mohammad Zuhair Khan
                        1 hour ago






                      • 1




                        @MohammadZuhairKhan Yes; just personally, I would mention that case, since often people assume that youre working in $mathbb{R}$ instead of the extended number system. I would just put it to make sure there's no chance of getting points off.
                        – Ovi
                        1 hour ago










                      • Oh okay thank you. Ofcourse, as it is done already, I can not (and would not) update my answer. But is there any other solution for this problem? I have a suspicion that this might be an open problem, or atleast a sensible version of it will.
                        – Mohammad Zuhair Khan
                        59 mins ago










                      • @MohammadZuhairKhan I am thinking about it
                        – Ovi
                        57 mins ago














                      1












                      1








                      1






                      Good try; but there's something that needs fixing. When $n$ is composite, the product diverges to $infty$; so you should define $text {sgn} (infty) = 1$.






                      share|cite|improve this answer












                      Good try; but there's something that needs fixing. When $n$ is composite, the product diverges to $infty$; so you should define $text {sgn} (infty) = 1$.







                      share|cite|improve this answer












                      share|cite|improve this answer



                      share|cite|improve this answer










                      answered 1 hour ago









                      Ovi

                      12.2k1038109




                      12.2k1038109












                      • Thanks for the advice. But, unless I am mistaken, $+infty gt 0?$
                        – Mohammad Zuhair Khan
                        1 hour ago






                      • 1




                        @MohammadZuhairKhan Yes; just personally, I would mention that case, since often people assume that youre working in $mathbb{R}$ instead of the extended number system. I would just put it to make sure there's no chance of getting points off.
                        – Ovi
                        1 hour ago










                      • Oh okay thank you. Ofcourse, as it is done already, I can not (and would not) update my answer. But is there any other solution for this problem? I have a suspicion that this might be an open problem, or atleast a sensible version of it will.
                        – Mohammad Zuhair Khan
                        59 mins ago










                      • @MohammadZuhairKhan I am thinking about it
                        – Ovi
                        57 mins ago


















                      • Thanks for the advice. But, unless I am mistaken, $+infty gt 0?$
                        – Mohammad Zuhair Khan
                        1 hour ago






                      • 1




                        @MohammadZuhairKhan Yes; just personally, I would mention that case, since often people assume that youre working in $mathbb{R}$ instead of the extended number system. I would just put it to make sure there's no chance of getting points off.
                        – Ovi
                        1 hour ago










                      • Oh okay thank you. Ofcourse, as it is done already, I can not (and would not) update my answer. But is there any other solution for this problem? I have a suspicion that this might be an open problem, or atleast a sensible version of it will.
                        – Mohammad Zuhair Khan
                        59 mins ago










                      • @MohammadZuhairKhan I am thinking about it
                        – Ovi
                        57 mins ago
















                      Thanks for the advice. But, unless I am mistaken, $+infty gt 0?$
                      – Mohammad Zuhair Khan
                      1 hour ago




                      Thanks for the advice. But, unless I am mistaken, $+infty gt 0?$
                      – Mohammad Zuhair Khan
                      1 hour ago




                      1




                      1




                      @MohammadZuhairKhan Yes; just personally, I would mention that case, since often people assume that youre working in $mathbb{R}$ instead of the extended number system. I would just put it to make sure there's no chance of getting points off.
                      – Ovi
                      1 hour ago




                      @MohammadZuhairKhan Yes; just personally, I would mention that case, since often people assume that youre working in $mathbb{R}$ instead of the extended number system. I would just put it to make sure there's no chance of getting points off.
                      – Ovi
                      1 hour ago












                      Oh okay thank you. Ofcourse, as it is done already, I can not (and would not) update my answer. But is there any other solution for this problem? I have a suspicion that this might be an open problem, or atleast a sensible version of it will.
                      – Mohammad Zuhair Khan
                      59 mins ago




                      Oh okay thank you. Ofcourse, as it is done already, I can not (and would not) update my answer. But is there any other solution for this problem? I have a suspicion that this might be an open problem, or atleast a sensible version of it will.
                      – Mohammad Zuhair Khan
                      59 mins ago












                      @MohammadZuhairKhan I am thinking about it
                      – Ovi
                      57 mins ago




                      @MohammadZuhairKhan I am thinking about it
                      – Ovi
                      57 mins ago











                      1














                      Instead of the sign function, you could potentially use the Kronecker delta, which is defined as



                      $$delta_{mn}=begin{cases}
                      1 & text{if }n=m\
                      0 & text{if }nneq m
                      end{cases}$$



                      It basically compares two numbers and gives $1$ if there is a match and $0$ otherwise. By summing over such Kronecker delta's, you could build:



                      $$theta(n)=nsum_{i=1}^{infty}delta_{np_i}$$



                      (where $p_i$ is the $i$-th prime number). However, I'm wondering if this would be accepted because it kind of bypasses the question of "checking if $n$ is prime". Both our formulas are just a nice rewording of the "text-form" formula given in the question, so I'm not certain this was the kind of answer that was expected.






                      share|cite|improve this answer


























                        1














                        Instead of the sign function, you could potentially use the Kronecker delta, which is defined as



                        $$delta_{mn}=begin{cases}
                        1 & text{if }n=m\
                        0 & text{if }nneq m
                        end{cases}$$



                        It basically compares two numbers and gives $1$ if there is a match and $0$ otherwise. By summing over such Kronecker delta's, you could build:



                        $$theta(n)=nsum_{i=1}^{infty}delta_{np_i}$$



                        (where $p_i$ is the $i$-th prime number). However, I'm wondering if this would be accepted because it kind of bypasses the question of "checking if $n$ is prime". Both our formulas are just a nice rewording of the "text-form" formula given in the question, so I'm not certain this was the kind of answer that was expected.






                        share|cite|improve this answer
























                          1












                          1








                          1






                          Instead of the sign function, you could potentially use the Kronecker delta, which is defined as



                          $$delta_{mn}=begin{cases}
                          1 & text{if }n=m\
                          0 & text{if }nneq m
                          end{cases}$$



                          It basically compares two numbers and gives $1$ if there is a match and $0$ otherwise. By summing over such Kronecker delta's, you could build:



                          $$theta(n)=nsum_{i=1}^{infty}delta_{np_i}$$



                          (where $p_i$ is the $i$-th prime number). However, I'm wondering if this would be accepted because it kind of bypasses the question of "checking if $n$ is prime". Both our formulas are just a nice rewording of the "text-form" formula given in the question, so I'm not certain this was the kind of answer that was expected.






                          share|cite|improve this answer












                          Instead of the sign function, you could potentially use the Kronecker delta, which is defined as



                          $$delta_{mn}=begin{cases}
                          1 & text{if }n=m\
                          0 & text{if }nneq m
                          end{cases}$$



                          It basically compares two numbers and gives $1$ if there is a match and $0$ otherwise. By summing over such Kronecker delta's, you could build:



                          $$theta(n)=nsum_{i=1}^{infty}delta_{np_i}$$



                          (where $p_i$ is the $i$-th prime number). However, I'm wondering if this would be accepted because it kind of bypasses the question of "checking if $n$ is prime". Both our formulas are just a nice rewording of the "text-form" formula given in the question, so I'm not certain this was the kind of answer that was expected.







                          share|cite|improve this answer












                          share|cite|improve this answer



                          share|cite|improve this answer










                          answered 30 mins ago









                          orion2112

                          436210




                          436210






























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