Kaisa Matomäki, Maksym Radziwill, Xuancheng Shao, Joni Teräväinen and I have just uploaded our preprint “Singmaster’s conjecture in the interior of Pascal’s Triangle” to the arXiv. This paper uses the theory of exponential sums over prime numbers to advance a well-known conjecture by Singmaster that claims that any natural number is greater than ${1}$ there is at most a limited number in Pascal’s triangle. That is, for any whole number ${t geq 2}$, there are at most ${O (1)}$ Solutions to the equation

$displaystyle binom {n} {m} = t (1)$

With”https://s0.wp.com/latex.php?latex=%7B1+%5Cleq+m+%3C+n%7D&bg=ffffff&fg=000000&s=0&c=20201002″ old =”{1 leqm < n}" class="latex" />. Currently, the largest number of solutions known to be achievable are eight, with ${t}$ equals

$displaystyle 3003 = binom {3003} {1} = binom {78} {2} = binom {15} {5} = binom {14} {6} = binom {14} {8} = binomial {15} {10}$

$displaystyle = binom {78} {76} = binom {3003} {3002}.$

Because of the symmetry ${ binom {n} {m} = binom {n} {nm}}$ of Pascal’s triangle, it is natural to focus attention on the left half ${1 leq m leq n / 2}$ of the triangle.

Our main result defines this assumption in the “inner” area of â€‹â€‹the triangle:

Movement 1 (Singmeister conjecture inside the triangle) If”https://s0.wp.com/latex.php?latex=%7B0+%3C+%5Cvarepsilon+%3C+1%7D&bg=ffffff&fg=000000&s=0&c=20201002″ old =”{0 < varepsilon < 1}" class="latex" /> and ${t}$ is big enough depending on ${ varepsilon}$, there are at most two solutions for (1) in the region

$displaystyle exp ( log ^ {2/3 + varepsilon} n) leq m leq n / 2 (2)$

and thus a maximum of four in the region

$displaystyle exp ( log ^ {2/3 + varepsilon} n) leq m leq n - exp ( log ^ {2/3 + varepsilon} n).$

In addition, there is at most one solution in the region

$displaystyle exp ( log ^ {2/3 + varepsilon} n) leq m leq n / exp ( log ^ {1- varepsilon} n).$

In order to fully verify Singmaster’s guess, in view of this result it is sufficient to verify the guess in the edge area

(or equivalent”https://s0.wp.com/latex.php?latex=%7Bn+-+%5Cexp%28%5Clog%5E%7B2%2F3%2B%5Cvarepsilon%7D+n%29+%3C+m+%5Cleq+ n% 7D & bg = ffffff & fg = 000000 & s = 0 & c = 20201002″ old =”{n – exp ( log ^ {2/3 + varepsilon} n) < m leq n}" class="latex" />). The upper bound of two for the number of solutions in region (7) is due to the infinite solution family of the equation. best possible

$displaystyle binom {n + 1} {m + 1} = binom {n} {m + 2} (3)$

come from ${n = F_ {2j + 2} F_ {2j + 3} -1}$, ${m = F_ {2j} F_ {2j + 3} -1}$ and ${F_j}$ is the ${j ^ {th}}$ Fibonacci number.

The look of the crowd ${ exp ( log ^ {2/3 + varepsilon} n)}$ in Theorem 1, readers may be familiar with Vinogradov’s bounds on exponential sums, which are ultimately the new main component of our argument. In principle, this threshold could be lowered if we had stronger limits for exponential sums.

To try to control solutions for (1), we use a combination of an “Archimedean” and a “non-Archimedean” approach. In the “Archimedean” approach (based on an earlier work by Kane on this problem) we consider ${n, m}$ mainly as real numbers rather than integers, and in terms of the gamma function, express (1) as

$displaystyle frac { Gamma (n + 1)} { Gamma (m + 1) Gamma (n-m + 1)} = t.$

This equation can be used to solve for ${n}$ in terms of ${m, t}$ how

$displaystyle n = f_t (m)$

for a certain real analytic function ${f_t}$ whose asymptotics are easy to calculate (for example one has the asymptotics ${f_t (m) asymp mt ^ {1 / m}}$). One can then see the problem by trying to control the number of grid points on the graph ${ {(m, f_t (m)): m in { bf R} }}$. Here we can take advantage of the fact that in the regime ${m leq f_t (m) / 2}$ (corresponds to the work in the left half ${m leq n / 2}$ of Pascal’s triangle), the function ${f_t}$ can be shown as convex, but not too convex, in the sense that one has both upper and lower bounds on the second derivative of ${f_t}$ (actually you can show that ${f '' _ t (m) asymp f_t (m) ( log t / m ^ 2) ^ 2}$). This can be used to rule out the possibility of having a cluster of three or more neighboring grid points on the graph ${ {(m, f_t (m)): m in { bf R} }}$, basically because the area bounded by the triangle connecting three of these points is between ${0}$ and ${1/2}$which contradicts Pick’s theorem. In developing these ideas, we were able to show

Suggestion 2 To let ${ varepsilon> 0}” class=”latex” />, and assume is big enough depending on ${ varepsilon}$. If ${(m, n)}$ is a solution of (1) in the left half ${m leq n / 2}$ of Pascal’s triangle, then there is at most one other solution ${(m ', n')}$ to this equation in the left half with

$displaystyle | m-m '| + | n-n '| ll exp (( log log t) ^ {1- varepsilon}).$

Here, too, the example from (3) shows that a cluster of two solutions is entirely possible; the convexity argument only takes effect when you have a cluster of three or more solutions.

To complete the proof of Theorem 1, one has to show that any two solutions ${(m, n), (m ', n')}$ to (1) in the region of interest must be close enough for the above suggestion to apply. Here we switch to the “non-Archimedean” approach, in which we use the ${p}$-adic ratings ${ nu_p ( binom {n} {m})}$ the binomial coefficient, defined as the frequency of a prime number ${p}$ Splits ${ binom {n} {m}}$. From the fundamental theorem of arithmetic, there is a collision

$displaystyle binom {n} {m} = binom {n '} {m'}$

between binomial coefficients occurs if and only if there is a match between the ratings

$displaystyle nu_p ( binom {n} {m}) = nu_p ( binom {n '} {m'}). (4)$

From the Legendre formula

$displaystyle nu_p (n!) = sum_ {j = 1} ^ infty lfloor frac {n} {p ^ j} rfloor$

we can rewrite this latter identity (4) as

$displaystyle sum_ {j = 1} ^ infty { frac {m} {p ^ j} } + { frac {nm} {p ^ j} } - { frac {n} { p ^ j} } = sum_ {j = 1} ^ infty { frac {m '} {p ^ j} } + { frac {n'-m'} {p ^ j} } - { frac {n '} {p ^ j} }, (5)$

Where ${ {x }: = x - lfloor x rfloor}$ denotes the fraction of ${x}$. (These sums are not really infinite because the summands vanish once ${p ^ j}$ is bigger than ${ max (n, n ')}$.)

A key idea in our approach is to look at this condition (5) statistical, for example by looking ${p}$ as a randomly drawn prime number from an interval such as such ${[P, P + P log^{-100} P]}$ for some suitably chosen scale parameters ${P}$so that the two sides of (5) now become random variables. It is then advantageous to compare correlations between these two random variables and some additional test random variables. For example when ${n}$ and ${n '}$ are far apart, then one would expect the left hand side of (5) to have a higher correlation with the fraction ${ { frac {n} {p} }}$because this term appears in the summation on the left, but not on the right. Similar if ${m}$ and ${m '}$ are far apart (although there are some annoying cases that need to be dealt with separately in the event of an “unexpected adequacy”, for example when ${n'-m '}$ is a rational multiple of ${m}$ where the rational has bounded numerators and denominators). To carry out this strategy, it turns out (after some standard Fourier expansion) that you have good control over exponential sums such as

$displaystyle sum_ {P leq p leq P + log ^ {- 100} P} e ( frac {N} {p} + frac {M} {p ^ j})$

for different parameter selection ${P, N, M, j}$, Where ${e ( theta): = e ^ {2 pi i theta}}$. Fortunately, Vinogradov’s methods (the more general with sums like ${ sum_ {n in I} e (f (n))}$ and ${ sum_ {p in I} e (f (p))}$ for various analysis functions ${f}$) can give useful limits on such sums as long as ${N}$ and ${M}$ aren’t too big compared to ${P}$; more precisely, Vinogradov’s estimates are not trivial in the regime ${N, M ll exp ( log ^ {3 / 2- varepsilon} P)}$, and this ultimately leads to a distance barrier

$displaystyle m '- m ll_ varepsilon exp ( log ^ {2/3 + varepsilon} (n + n'))$

between each colliding pair ${(n, m), (n ', m')}$ bound in the left half of Pascal’s triangle, as well as the variant

$displaystyle n '- n ll_ varepsilon exp ( log ^ {2/3 + varepsilon} (n + n'))$

$displaystyle m ', m geq exp ( log ^ {2/3 + varepsilon} (n + n')).$

Compare these limits to Theorem 2 and use some basic guesses about the function ${f_t}$, we can close sentence 1.

Modifying the arguments also gives similar results for the equation

$displaystyle (n) _m = t (6)$

Where ${(n) _m: = n (n-1) dots (n-m + 1)}$ is the decreasing factorial:

Sentence 3 If”https://s0.wp.com/latex.php?latex=%7B0+%3C+%5Cvarepsilon+%3C+1%7D&bg=ffffff&fg=000000&s=0&c=20201002″ old =”{0 < varepsilon < 1}" class="latex" /> and ${t}$ is big enough depending on ${ varepsilon}$, there are at most two solutions for (6) in the region

Again, the upper limit of two is thanks to identities like

$displaystyle (a ^ 2-a) _ {a ^ 2-2a} = (a ^ 2-a-1) _ {a ^ 2-2a + 1}.$