Degree of polynomial interpolating the primes











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down vote

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The polynomial $p_3(x)$ passes through the points
$(1,2), (2,3), (3,5)$, where $2,3,5$ are the first three primes:
$$
p_3(x) = frac{x^2}{2}-frac{x}{2}+2 ;.
$$

Similarly, one can form an interpolating polynomial $p_n(x)$ that
passes through the first $n$ primes.
For example:
$$
p_5(x) = frac{x^4}{8}-frac{17
x^3}{12}+frac{47
x^2}{8}-frac{103 x}{12}+6 ;.
$$

One can check that
begin{eqnarray}
p_5(1) &=& 2 \
p_5(2) &=& 3 \
p_5(3) &=& 5 \
p_5(4) &=& 7 \
p_5(5) &=& 11 ;.
end{eqnarray}



My question is:




Q. Is the degree of $p_n(x)$ ever strictly less than $n{-}1$, for any $n$?




The answer to Q is positive if a "coincidence" occurs,
such that a smaller degree
polynomial captures those $n$ prime points.
Do such coincidences ever occur?










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  • 1




    See also math.stackexchange.com/questions/577448/…
    – lhf
    Nov 22 at 18:00






  • 1




    It is for $nle 200$.
    – lhf
    Nov 22 at 18:32






  • 1




    The edit helps, but the question might be clearer still if it were phrased as "Is the degree of $p_n(x)$ ever strictly less than $n-1$ for any $n$?", in which case an affirmative answer would constitute a coincidence I know a few people (myself included) misunderstood the question at first.
    – mweiss
    Nov 25 at 1:25






  • 1




    @mweiss: Thanks, I followed your suggestion.
    – Joseph O'Rourke
    Nov 25 at 1:47






  • 1




    Interesting related fact, though probably not helpful towards an answer: The sum of the coefficients of $p_n(x)$ appears to converge. oeis.org/A092894
    – Steve Kass
    Nov 25 at 2:13















up vote
10
down vote

favorite
3












The polynomial $p_3(x)$ passes through the points
$(1,2), (2,3), (3,5)$, where $2,3,5$ are the first three primes:
$$
p_3(x) = frac{x^2}{2}-frac{x}{2}+2 ;.
$$

Similarly, one can form an interpolating polynomial $p_n(x)$ that
passes through the first $n$ primes.
For example:
$$
p_5(x) = frac{x^4}{8}-frac{17
x^3}{12}+frac{47
x^2}{8}-frac{103 x}{12}+6 ;.
$$

One can check that
begin{eqnarray}
p_5(1) &=& 2 \
p_5(2) &=& 3 \
p_5(3) &=& 5 \
p_5(4) &=& 7 \
p_5(5) &=& 11 ;.
end{eqnarray}



My question is:




Q. Is the degree of $p_n(x)$ ever strictly less than $n{-}1$, for any $n$?




The answer to Q is positive if a "coincidence" occurs,
such that a smaller degree
polynomial captures those $n$ prime points.
Do such coincidences ever occur?










share|cite|improve this question




















  • 1




    See also math.stackexchange.com/questions/577448/…
    – lhf
    Nov 22 at 18:00






  • 1




    It is for $nle 200$.
    – lhf
    Nov 22 at 18:32






  • 1




    The edit helps, but the question might be clearer still if it were phrased as "Is the degree of $p_n(x)$ ever strictly less than $n-1$ for any $n$?", in which case an affirmative answer would constitute a coincidence I know a few people (myself included) misunderstood the question at first.
    – mweiss
    Nov 25 at 1:25






  • 1




    @mweiss: Thanks, I followed your suggestion.
    – Joseph O'Rourke
    Nov 25 at 1:47






  • 1




    Interesting related fact, though probably not helpful towards an answer: The sum of the coefficients of $p_n(x)$ appears to converge. oeis.org/A092894
    – Steve Kass
    Nov 25 at 2:13













up vote
10
down vote

favorite
3









up vote
10
down vote

favorite
3






3





The polynomial $p_3(x)$ passes through the points
$(1,2), (2,3), (3,5)$, where $2,3,5$ are the first three primes:
$$
p_3(x) = frac{x^2}{2}-frac{x}{2}+2 ;.
$$

Similarly, one can form an interpolating polynomial $p_n(x)$ that
passes through the first $n$ primes.
For example:
$$
p_5(x) = frac{x^4}{8}-frac{17
x^3}{12}+frac{47
x^2}{8}-frac{103 x}{12}+6 ;.
$$

One can check that
begin{eqnarray}
p_5(1) &=& 2 \
p_5(2) &=& 3 \
p_5(3) &=& 5 \
p_5(4) &=& 7 \
p_5(5) &=& 11 ;.
end{eqnarray}



My question is:




Q. Is the degree of $p_n(x)$ ever strictly less than $n{-}1$, for any $n$?




The answer to Q is positive if a "coincidence" occurs,
such that a smaller degree
polynomial captures those $n$ prime points.
Do such coincidences ever occur?










share|cite|improve this question















The polynomial $p_3(x)$ passes through the points
$(1,2), (2,3), (3,5)$, where $2,3,5$ are the first three primes:
$$
p_3(x) = frac{x^2}{2}-frac{x}{2}+2 ;.
$$

Similarly, one can form an interpolating polynomial $p_n(x)$ that
passes through the first $n$ primes.
For example:
$$
p_5(x) = frac{x^4}{8}-frac{17
x^3}{12}+frac{47
x^2}{8}-frac{103 x}{12}+6 ;.
$$

One can check that
begin{eqnarray}
p_5(1) &=& 2 \
p_5(2) &=& 3 \
p_5(3) &=& 5 \
p_5(4) &=& 7 \
p_5(5) &=& 11 ;.
end{eqnarray}



My question is:




Q. Is the degree of $p_n(x)$ ever strictly less than $n{-}1$, for any $n$?




The answer to Q is positive if a "coincidence" occurs,
such that a smaller degree
polynomial captures those $n$ prime points.
Do such coincidences ever occur?







number-theory polynomials prime-numbers






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share|cite|improve this question













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share|cite|improve this question








edited Nov 25 at 1:46

























asked Nov 22 at 13:47









Joseph O'Rourke

17.5k348106




17.5k348106








  • 1




    See also math.stackexchange.com/questions/577448/…
    – lhf
    Nov 22 at 18:00






  • 1




    It is for $nle 200$.
    – lhf
    Nov 22 at 18:32






  • 1




    The edit helps, but the question might be clearer still if it were phrased as "Is the degree of $p_n(x)$ ever strictly less than $n-1$ for any $n$?", in which case an affirmative answer would constitute a coincidence I know a few people (myself included) misunderstood the question at first.
    – mweiss
    Nov 25 at 1:25






  • 1




    @mweiss: Thanks, I followed your suggestion.
    – Joseph O'Rourke
    Nov 25 at 1:47






  • 1




    Interesting related fact, though probably not helpful towards an answer: The sum of the coefficients of $p_n(x)$ appears to converge. oeis.org/A092894
    – Steve Kass
    Nov 25 at 2:13














  • 1




    See also math.stackexchange.com/questions/577448/…
    – lhf
    Nov 22 at 18:00






  • 1




    It is for $nle 200$.
    – lhf
    Nov 22 at 18:32






  • 1




    The edit helps, but the question might be clearer still if it were phrased as "Is the degree of $p_n(x)$ ever strictly less than $n-1$ for any $n$?", in which case an affirmative answer would constitute a coincidence I know a few people (myself included) misunderstood the question at first.
    – mweiss
    Nov 25 at 1:25






  • 1




    @mweiss: Thanks, I followed your suggestion.
    – Joseph O'Rourke
    Nov 25 at 1:47






  • 1




    Interesting related fact, though probably not helpful towards an answer: The sum of the coefficients of $p_n(x)$ appears to converge. oeis.org/A092894
    – Steve Kass
    Nov 25 at 2:13








1




1




See also math.stackexchange.com/questions/577448/…
– lhf
Nov 22 at 18:00




See also math.stackexchange.com/questions/577448/…
– lhf
Nov 22 at 18:00




1




1




It is for $nle 200$.
– lhf
Nov 22 at 18:32




It is for $nle 200$.
– lhf
Nov 22 at 18:32




1




1




The edit helps, but the question might be clearer still if it were phrased as "Is the degree of $p_n(x)$ ever strictly less than $n-1$ for any $n$?", in which case an affirmative answer would constitute a coincidence I know a few people (myself included) misunderstood the question at first.
– mweiss
Nov 25 at 1:25




The edit helps, but the question might be clearer still if it were phrased as "Is the degree of $p_n(x)$ ever strictly less than $n-1$ for any $n$?", in which case an affirmative answer would constitute a coincidence I know a few people (myself included) misunderstood the question at first.
– mweiss
Nov 25 at 1:25




1




1




@mweiss: Thanks, I followed your suggestion.
– Joseph O'Rourke
Nov 25 at 1:47




@mweiss: Thanks, I followed your suggestion.
– Joseph O'Rourke
Nov 25 at 1:47




1




1




Interesting related fact, though probably not helpful towards an answer: The sum of the coefficients of $p_n(x)$ appears to converge. oeis.org/A092894
– Steve Kass
Nov 25 at 2:13




Interesting related fact, though probably not helpful towards an answer: The sum of the coefficients of $p_n(x)$ appears to converge. oeis.org/A092894
– Steve Kass
Nov 25 at 2:13










3 Answers
3






active

oldest

votes

















up vote
5
down vote



accepted










The degree of $p_n(x)$ is always $n-1$. The proof is by induction.



Note that $p_1(x) = 2$ has degree $0$. Now assume that $p_{n}(x)$ has degree $n-1$. We want to prove that $p_{n+1}(x)$ has degree $n$. Assume otherwise, so $p_{n+1}(x)$ also had degree at most $n-1$. Then since $p_{n+1}(x)$ and $p_n(x)$ agree on the first $n$ values, it must be the case that $p_{n+1}(x) = p_n(x)$. In particular, to obtain a contradiction, it suffices to show that



$$p_n(n+1) ne^{?} p_{n+1}.$$



In fact, we simply will prove that $p_n(n+1)$ is always even which does the job.



We can write down a formula for $p_n(x)$, namely



$$p_n(x) = sum_{i=1}^{n} p_i cdot
frac{(x-1)(x-2) ldots widehat{(x-i)} ldots (x - n)}{(i-1)(i-2)
ldots widehat{(i-i)} ldots (i - n)},$$



where the hat indicates the term is omitted. This is clearly a polynomial of degree at most $n-1$ and $p_n(i) = p_i$. (This is the general formula for Lagrange interpolation specialized to this case.)



Hence



$$begin{aligned} p_n(n+1) = & sum_{i=1}^{n} p_i cdotfrac{ n!/(n+1-i)}{(i-1)! (n-i)! (-1)^{n-i}}\
= & (-1)^{n-1} sum_{i=1}^{n} p_i cdot frac{ n!}{(i-1)! (n+1-i)!} (-1)^{i-1} \
= & (-1)^{n-1} sum_{i=1}^{n} p_i cdot binom{n}{i-1} (-1)^{i-1}\
= & (-1)^{n-1} sum_{i=0}^{n-1} p_{i+1} binom{n}{i} (-1)^iend{aligned}$$



Now we use the fact that, with the exception of $p_1 = 2$, the primes are all odd. It follows that



$$p_{n}(n+1) equiv sum_{i=1}^{n-1} (-1)^i binom{n}{i} mod 2.$$



But now



$$sum_{i=1}^{n-1} (-1)^i binom{n}{i}
= (1-1)^n - 1 - (-1)^n equiv 0 mod 2,$$



is even for $n > 0$, and hence $p_{n}(n+1)$ is even, and thus $ne p_{n+1}$, as desired.






share|cite|improve this answer





















  • One can simplify this somewhat by noting that we don't need Lagrange interpolation; the method of forward differences is sufficient here and tells us that $n!$ times the coefficient of $x^{n-1}$ in $p_n$ is $sum_{i=0}^{n-1}(-1)^i{n-1choose i}cdot p_{i+1}$ which, similar to your argument, turns out to be odd, so not zero.
    – Milo Brandt
    Nov 25 at 19:01












  • wonderful proof
    – Sandeep Silwal
    Nov 25 at 21:03










  • There's nothing difficult about Lagrange interpolation --- it's an identity that proves itself. It would take longer to write out an argument using discrete derivatives and end up basically being the same, so I disagree it would be any simplification.
    – Lorem Ipsum
    Nov 25 at 21:49


















up vote
0
down vote













Not really an answer, but consider a more general question:




Does the polynomial interpolating $n$ consecutive primes $p_{m+1},dots,p_{m+n}$ always have maximum degree $n-1$?




The answer is a strong no because $3,5,7$ and $251,257,263,269$ are consecutive primes in arithmetic progression.
Small examples are known for $3 le n le 6$. See Wikipedia.



The polynomial interpolating the four consecutive primes $17, 19, 23, 29$ has degree $2$, not $3$. So does the polynomial interpolating the four consecutive primes $p_{m+1},dots,p_{m+4}$ for $m in {6,10,12,17,21,48,57,68,69,74,84,90,103,110,115,121,122,126,131,172,181}$.






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  • So what? You have answered negatively to a more general question. What about the particular case of $p_1,dots,p_n$?
    – Federico
    Nov 22 at 18:01


















up vote
-1
down vote













Yes. Given $n$ unique data points, there is a unique polynomial of degree $n-1$ that interpolates the data. This is one of the first results in the interpolation section of any numerical analysis/methods course. Using data points constructed with primes is a specific case of this. If you write out the interpolation conditions, you'll see that this is equivalent to solving an $n times n$ linear system.






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  • 3




    It is possible that the coefficient(s) of the highest degree term(s) is/are $0$, which would mean that the degree could be less than $n-1$. Actually the standard result is: There is a unique polynomial of degree at most $n-1$ interpolating $n$ points with different $x$-coordinates.
    – paw88789
    Nov 22 at 14:45











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3 Answers
3






active

oldest

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3 Answers
3






active

oldest

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active

oldest

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active

oldest

votes








up vote
5
down vote



accepted










The degree of $p_n(x)$ is always $n-1$. The proof is by induction.



Note that $p_1(x) = 2$ has degree $0$. Now assume that $p_{n}(x)$ has degree $n-1$. We want to prove that $p_{n+1}(x)$ has degree $n$. Assume otherwise, so $p_{n+1}(x)$ also had degree at most $n-1$. Then since $p_{n+1}(x)$ and $p_n(x)$ agree on the first $n$ values, it must be the case that $p_{n+1}(x) = p_n(x)$. In particular, to obtain a contradiction, it suffices to show that



$$p_n(n+1) ne^{?} p_{n+1}.$$



In fact, we simply will prove that $p_n(n+1)$ is always even which does the job.



We can write down a formula for $p_n(x)$, namely



$$p_n(x) = sum_{i=1}^{n} p_i cdot
frac{(x-1)(x-2) ldots widehat{(x-i)} ldots (x - n)}{(i-1)(i-2)
ldots widehat{(i-i)} ldots (i - n)},$$



where the hat indicates the term is omitted. This is clearly a polynomial of degree at most $n-1$ and $p_n(i) = p_i$. (This is the general formula for Lagrange interpolation specialized to this case.)



Hence



$$begin{aligned} p_n(n+1) = & sum_{i=1}^{n} p_i cdotfrac{ n!/(n+1-i)}{(i-1)! (n-i)! (-1)^{n-i}}\
= & (-1)^{n-1} sum_{i=1}^{n} p_i cdot frac{ n!}{(i-1)! (n+1-i)!} (-1)^{i-1} \
= & (-1)^{n-1} sum_{i=1}^{n} p_i cdot binom{n}{i-1} (-1)^{i-1}\
= & (-1)^{n-1} sum_{i=0}^{n-1} p_{i+1} binom{n}{i} (-1)^iend{aligned}$$



Now we use the fact that, with the exception of $p_1 = 2$, the primes are all odd. It follows that



$$p_{n}(n+1) equiv sum_{i=1}^{n-1} (-1)^i binom{n}{i} mod 2.$$



But now



$$sum_{i=1}^{n-1} (-1)^i binom{n}{i}
= (1-1)^n - 1 - (-1)^n equiv 0 mod 2,$$



is even for $n > 0$, and hence $p_{n}(n+1)$ is even, and thus $ne p_{n+1}$, as desired.






share|cite|improve this answer





















  • One can simplify this somewhat by noting that we don't need Lagrange interpolation; the method of forward differences is sufficient here and tells us that $n!$ times the coefficient of $x^{n-1}$ in $p_n$ is $sum_{i=0}^{n-1}(-1)^i{n-1choose i}cdot p_{i+1}$ which, similar to your argument, turns out to be odd, so not zero.
    – Milo Brandt
    Nov 25 at 19:01












  • wonderful proof
    – Sandeep Silwal
    Nov 25 at 21:03










  • There's nothing difficult about Lagrange interpolation --- it's an identity that proves itself. It would take longer to write out an argument using discrete derivatives and end up basically being the same, so I disagree it would be any simplification.
    – Lorem Ipsum
    Nov 25 at 21:49















up vote
5
down vote



accepted










The degree of $p_n(x)$ is always $n-1$. The proof is by induction.



Note that $p_1(x) = 2$ has degree $0$. Now assume that $p_{n}(x)$ has degree $n-1$. We want to prove that $p_{n+1}(x)$ has degree $n$. Assume otherwise, so $p_{n+1}(x)$ also had degree at most $n-1$. Then since $p_{n+1}(x)$ and $p_n(x)$ agree on the first $n$ values, it must be the case that $p_{n+1}(x) = p_n(x)$. In particular, to obtain a contradiction, it suffices to show that



$$p_n(n+1) ne^{?} p_{n+1}.$$



In fact, we simply will prove that $p_n(n+1)$ is always even which does the job.



We can write down a formula for $p_n(x)$, namely



$$p_n(x) = sum_{i=1}^{n} p_i cdot
frac{(x-1)(x-2) ldots widehat{(x-i)} ldots (x - n)}{(i-1)(i-2)
ldots widehat{(i-i)} ldots (i - n)},$$



where the hat indicates the term is omitted. This is clearly a polynomial of degree at most $n-1$ and $p_n(i) = p_i$. (This is the general formula for Lagrange interpolation specialized to this case.)



Hence



$$begin{aligned} p_n(n+1) = & sum_{i=1}^{n} p_i cdotfrac{ n!/(n+1-i)}{(i-1)! (n-i)! (-1)^{n-i}}\
= & (-1)^{n-1} sum_{i=1}^{n} p_i cdot frac{ n!}{(i-1)! (n+1-i)!} (-1)^{i-1} \
= & (-1)^{n-1} sum_{i=1}^{n} p_i cdot binom{n}{i-1} (-1)^{i-1}\
= & (-1)^{n-1} sum_{i=0}^{n-1} p_{i+1} binom{n}{i} (-1)^iend{aligned}$$



Now we use the fact that, with the exception of $p_1 = 2$, the primes are all odd. It follows that



$$p_{n}(n+1) equiv sum_{i=1}^{n-1} (-1)^i binom{n}{i} mod 2.$$



But now



$$sum_{i=1}^{n-1} (-1)^i binom{n}{i}
= (1-1)^n - 1 - (-1)^n equiv 0 mod 2,$$



is even for $n > 0$, and hence $p_{n}(n+1)$ is even, and thus $ne p_{n+1}$, as desired.






share|cite|improve this answer





















  • One can simplify this somewhat by noting that we don't need Lagrange interpolation; the method of forward differences is sufficient here and tells us that $n!$ times the coefficient of $x^{n-1}$ in $p_n$ is $sum_{i=0}^{n-1}(-1)^i{n-1choose i}cdot p_{i+1}$ which, similar to your argument, turns out to be odd, so not zero.
    – Milo Brandt
    Nov 25 at 19:01












  • wonderful proof
    – Sandeep Silwal
    Nov 25 at 21:03










  • There's nothing difficult about Lagrange interpolation --- it's an identity that proves itself. It would take longer to write out an argument using discrete derivatives and end up basically being the same, so I disagree it would be any simplification.
    – Lorem Ipsum
    Nov 25 at 21:49













up vote
5
down vote



accepted







up vote
5
down vote



accepted






The degree of $p_n(x)$ is always $n-1$. The proof is by induction.



Note that $p_1(x) = 2$ has degree $0$. Now assume that $p_{n}(x)$ has degree $n-1$. We want to prove that $p_{n+1}(x)$ has degree $n$. Assume otherwise, so $p_{n+1}(x)$ also had degree at most $n-1$. Then since $p_{n+1}(x)$ and $p_n(x)$ agree on the first $n$ values, it must be the case that $p_{n+1}(x) = p_n(x)$. In particular, to obtain a contradiction, it suffices to show that



$$p_n(n+1) ne^{?} p_{n+1}.$$



In fact, we simply will prove that $p_n(n+1)$ is always even which does the job.



We can write down a formula for $p_n(x)$, namely



$$p_n(x) = sum_{i=1}^{n} p_i cdot
frac{(x-1)(x-2) ldots widehat{(x-i)} ldots (x - n)}{(i-1)(i-2)
ldots widehat{(i-i)} ldots (i - n)},$$



where the hat indicates the term is omitted. This is clearly a polynomial of degree at most $n-1$ and $p_n(i) = p_i$. (This is the general formula for Lagrange interpolation specialized to this case.)



Hence



$$begin{aligned} p_n(n+1) = & sum_{i=1}^{n} p_i cdotfrac{ n!/(n+1-i)}{(i-1)! (n-i)! (-1)^{n-i}}\
= & (-1)^{n-1} sum_{i=1}^{n} p_i cdot frac{ n!}{(i-1)! (n+1-i)!} (-1)^{i-1} \
= & (-1)^{n-1} sum_{i=1}^{n} p_i cdot binom{n}{i-1} (-1)^{i-1}\
= & (-1)^{n-1} sum_{i=0}^{n-1} p_{i+1} binom{n}{i} (-1)^iend{aligned}$$



Now we use the fact that, with the exception of $p_1 = 2$, the primes are all odd. It follows that



$$p_{n}(n+1) equiv sum_{i=1}^{n-1} (-1)^i binom{n}{i} mod 2.$$



But now



$$sum_{i=1}^{n-1} (-1)^i binom{n}{i}
= (1-1)^n - 1 - (-1)^n equiv 0 mod 2,$$



is even for $n > 0$, and hence $p_{n}(n+1)$ is even, and thus $ne p_{n+1}$, as desired.






share|cite|improve this answer












The degree of $p_n(x)$ is always $n-1$. The proof is by induction.



Note that $p_1(x) = 2$ has degree $0$. Now assume that $p_{n}(x)$ has degree $n-1$. We want to prove that $p_{n+1}(x)$ has degree $n$. Assume otherwise, so $p_{n+1}(x)$ also had degree at most $n-1$. Then since $p_{n+1}(x)$ and $p_n(x)$ agree on the first $n$ values, it must be the case that $p_{n+1}(x) = p_n(x)$. In particular, to obtain a contradiction, it suffices to show that



$$p_n(n+1) ne^{?} p_{n+1}.$$



In fact, we simply will prove that $p_n(n+1)$ is always even which does the job.



We can write down a formula for $p_n(x)$, namely



$$p_n(x) = sum_{i=1}^{n} p_i cdot
frac{(x-1)(x-2) ldots widehat{(x-i)} ldots (x - n)}{(i-1)(i-2)
ldots widehat{(i-i)} ldots (i - n)},$$



where the hat indicates the term is omitted. This is clearly a polynomial of degree at most $n-1$ and $p_n(i) = p_i$. (This is the general formula for Lagrange interpolation specialized to this case.)



Hence



$$begin{aligned} p_n(n+1) = & sum_{i=1}^{n} p_i cdotfrac{ n!/(n+1-i)}{(i-1)! (n-i)! (-1)^{n-i}}\
= & (-1)^{n-1} sum_{i=1}^{n} p_i cdot frac{ n!}{(i-1)! (n+1-i)!} (-1)^{i-1} \
= & (-1)^{n-1} sum_{i=1}^{n} p_i cdot binom{n}{i-1} (-1)^{i-1}\
= & (-1)^{n-1} sum_{i=0}^{n-1} p_{i+1} binom{n}{i} (-1)^iend{aligned}$$



Now we use the fact that, with the exception of $p_1 = 2$, the primes are all odd. It follows that



$$p_{n}(n+1) equiv sum_{i=1}^{n-1} (-1)^i binom{n}{i} mod 2.$$



But now



$$sum_{i=1}^{n-1} (-1)^i binom{n}{i}
= (1-1)^n - 1 - (-1)^n equiv 0 mod 2,$$



is even for $n > 0$, and hence $p_{n}(n+1)$ is even, and thus $ne p_{n+1}$, as desired.







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answered Nov 25 at 18:48









Lorem Ipsum

1312




1312












  • One can simplify this somewhat by noting that we don't need Lagrange interpolation; the method of forward differences is sufficient here and tells us that $n!$ times the coefficient of $x^{n-1}$ in $p_n$ is $sum_{i=0}^{n-1}(-1)^i{n-1choose i}cdot p_{i+1}$ which, similar to your argument, turns out to be odd, so not zero.
    – Milo Brandt
    Nov 25 at 19:01












  • wonderful proof
    – Sandeep Silwal
    Nov 25 at 21:03










  • There's nothing difficult about Lagrange interpolation --- it's an identity that proves itself. It would take longer to write out an argument using discrete derivatives and end up basically being the same, so I disagree it would be any simplification.
    – Lorem Ipsum
    Nov 25 at 21:49


















  • One can simplify this somewhat by noting that we don't need Lagrange interpolation; the method of forward differences is sufficient here and tells us that $n!$ times the coefficient of $x^{n-1}$ in $p_n$ is $sum_{i=0}^{n-1}(-1)^i{n-1choose i}cdot p_{i+1}$ which, similar to your argument, turns out to be odd, so not zero.
    – Milo Brandt
    Nov 25 at 19:01












  • wonderful proof
    – Sandeep Silwal
    Nov 25 at 21:03










  • There's nothing difficult about Lagrange interpolation --- it's an identity that proves itself. It would take longer to write out an argument using discrete derivatives and end up basically being the same, so I disagree it would be any simplification.
    – Lorem Ipsum
    Nov 25 at 21:49
















One can simplify this somewhat by noting that we don't need Lagrange interpolation; the method of forward differences is sufficient here and tells us that $n!$ times the coefficient of $x^{n-1}$ in $p_n$ is $sum_{i=0}^{n-1}(-1)^i{n-1choose i}cdot p_{i+1}$ which, similar to your argument, turns out to be odd, so not zero.
– Milo Brandt
Nov 25 at 19:01






One can simplify this somewhat by noting that we don't need Lagrange interpolation; the method of forward differences is sufficient here and tells us that $n!$ times the coefficient of $x^{n-1}$ in $p_n$ is $sum_{i=0}^{n-1}(-1)^i{n-1choose i}cdot p_{i+1}$ which, similar to your argument, turns out to be odd, so not zero.
– Milo Brandt
Nov 25 at 19:01














wonderful proof
– Sandeep Silwal
Nov 25 at 21:03




wonderful proof
– Sandeep Silwal
Nov 25 at 21:03












There's nothing difficult about Lagrange interpolation --- it's an identity that proves itself. It would take longer to write out an argument using discrete derivatives and end up basically being the same, so I disagree it would be any simplification.
– Lorem Ipsum
Nov 25 at 21:49




There's nothing difficult about Lagrange interpolation --- it's an identity that proves itself. It would take longer to write out an argument using discrete derivatives and end up basically being the same, so I disagree it would be any simplification.
– Lorem Ipsum
Nov 25 at 21:49










up vote
0
down vote













Not really an answer, but consider a more general question:




Does the polynomial interpolating $n$ consecutive primes $p_{m+1},dots,p_{m+n}$ always have maximum degree $n-1$?




The answer is a strong no because $3,5,7$ and $251,257,263,269$ are consecutive primes in arithmetic progression.
Small examples are known for $3 le n le 6$. See Wikipedia.



The polynomial interpolating the four consecutive primes $17, 19, 23, 29$ has degree $2$, not $3$. So does the polynomial interpolating the four consecutive primes $p_{m+1},dots,p_{m+4}$ for $m in {6,10,12,17,21,48,57,68,69,74,84,90,103,110,115,121,122,126,131,172,181}$.






share|cite|improve this answer























  • So what? You have answered negatively to a more general question. What about the particular case of $p_1,dots,p_n$?
    – Federico
    Nov 22 at 18:01















up vote
0
down vote













Not really an answer, but consider a more general question:




Does the polynomial interpolating $n$ consecutive primes $p_{m+1},dots,p_{m+n}$ always have maximum degree $n-1$?




The answer is a strong no because $3,5,7$ and $251,257,263,269$ are consecutive primes in arithmetic progression.
Small examples are known for $3 le n le 6$. See Wikipedia.



The polynomial interpolating the four consecutive primes $17, 19, 23, 29$ has degree $2$, not $3$. So does the polynomial interpolating the four consecutive primes $p_{m+1},dots,p_{m+4}$ for $m in {6,10,12,17,21,48,57,68,69,74,84,90,103,110,115,121,122,126,131,172,181}$.






share|cite|improve this answer























  • So what? You have answered negatively to a more general question. What about the particular case of $p_1,dots,p_n$?
    – Federico
    Nov 22 at 18:01













up vote
0
down vote










up vote
0
down vote









Not really an answer, but consider a more general question:




Does the polynomial interpolating $n$ consecutive primes $p_{m+1},dots,p_{m+n}$ always have maximum degree $n-1$?




The answer is a strong no because $3,5,7$ and $251,257,263,269$ are consecutive primes in arithmetic progression.
Small examples are known for $3 le n le 6$. See Wikipedia.



The polynomial interpolating the four consecutive primes $17, 19, 23, 29$ has degree $2$, not $3$. So does the polynomial interpolating the four consecutive primes $p_{m+1},dots,p_{m+4}$ for $m in {6,10,12,17,21,48,57,68,69,74,84,90,103,110,115,121,122,126,131,172,181}$.






share|cite|improve this answer














Not really an answer, but consider a more general question:




Does the polynomial interpolating $n$ consecutive primes $p_{m+1},dots,p_{m+n}$ always have maximum degree $n-1$?




The answer is a strong no because $3,5,7$ and $251,257,263,269$ are consecutive primes in arithmetic progression.
Small examples are known for $3 le n le 6$. See Wikipedia.



The polynomial interpolating the four consecutive primes $17, 19, 23, 29$ has degree $2$, not $3$. So does the polynomial interpolating the four consecutive primes $p_{m+1},dots,p_{m+4}$ for $m in {6,10,12,17,21,48,57,68,69,74,84,90,103,110,115,121,122,126,131,172,181}$.







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited Nov 23 at 16:32

























answered Nov 22 at 17:38









lhf

161k9165384




161k9165384












  • So what? You have answered negatively to a more general question. What about the particular case of $p_1,dots,p_n$?
    – Federico
    Nov 22 at 18:01


















  • So what? You have answered negatively to a more general question. What about the particular case of $p_1,dots,p_n$?
    – Federico
    Nov 22 at 18:01
















So what? You have answered negatively to a more general question. What about the particular case of $p_1,dots,p_n$?
– Federico
Nov 22 at 18:01




So what? You have answered negatively to a more general question. What about the particular case of $p_1,dots,p_n$?
– Federico
Nov 22 at 18:01










up vote
-1
down vote













Yes. Given $n$ unique data points, there is a unique polynomial of degree $n-1$ that interpolates the data. This is one of the first results in the interpolation section of any numerical analysis/methods course. Using data points constructed with primes is a specific case of this. If you write out the interpolation conditions, you'll see that this is equivalent to solving an $n times n$ linear system.






share|cite|improve this answer

















  • 3




    It is possible that the coefficient(s) of the highest degree term(s) is/are $0$, which would mean that the degree could be less than $n-1$. Actually the standard result is: There is a unique polynomial of degree at most $n-1$ interpolating $n$ points with different $x$-coordinates.
    – paw88789
    Nov 22 at 14:45















up vote
-1
down vote













Yes. Given $n$ unique data points, there is a unique polynomial of degree $n-1$ that interpolates the data. This is one of the first results in the interpolation section of any numerical analysis/methods course. Using data points constructed with primes is a specific case of this. If you write out the interpolation conditions, you'll see that this is equivalent to solving an $n times n$ linear system.






share|cite|improve this answer

















  • 3




    It is possible that the coefficient(s) of the highest degree term(s) is/are $0$, which would mean that the degree could be less than $n-1$. Actually the standard result is: There is a unique polynomial of degree at most $n-1$ interpolating $n$ points with different $x$-coordinates.
    – paw88789
    Nov 22 at 14:45













up vote
-1
down vote










up vote
-1
down vote









Yes. Given $n$ unique data points, there is a unique polynomial of degree $n-1$ that interpolates the data. This is one of the first results in the interpolation section of any numerical analysis/methods course. Using data points constructed with primes is a specific case of this. If you write out the interpolation conditions, you'll see that this is equivalent to solving an $n times n$ linear system.






share|cite|improve this answer












Yes. Given $n$ unique data points, there is a unique polynomial of degree $n-1$ that interpolates the data. This is one of the first results in the interpolation section of any numerical analysis/methods course. Using data points constructed with primes is a specific case of this. If you write out the interpolation conditions, you'll see that this is equivalent to solving an $n times n$ linear system.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Nov 22 at 14:30









Eric

11




11








  • 3




    It is possible that the coefficient(s) of the highest degree term(s) is/are $0$, which would mean that the degree could be less than $n-1$. Actually the standard result is: There is a unique polynomial of degree at most $n-1$ interpolating $n$ points with different $x$-coordinates.
    – paw88789
    Nov 22 at 14:45














  • 3




    It is possible that the coefficient(s) of the highest degree term(s) is/are $0$, which would mean that the degree could be less than $n-1$. Actually the standard result is: There is a unique polynomial of degree at most $n-1$ interpolating $n$ points with different $x$-coordinates.
    – paw88789
    Nov 22 at 14:45








3




3




It is possible that the coefficient(s) of the highest degree term(s) is/are $0$, which would mean that the degree could be less than $n-1$. Actually the standard result is: There is a unique polynomial of degree at most $n-1$ interpolating $n$ points with different $x$-coordinates.
– paw88789
Nov 22 at 14:45




It is possible that the coefficient(s) of the highest degree term(s) is/are $0$, which would mean that the degree could be less than $n-1$. Actually the standard result is: There is a unique polynomial of degree at most $n-1$ interpolating $n$ points with different $x$-coordinates.
– paw88789
Nov 22 at 14:45


















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