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In combinatorial number theory, it is clear that the only number that appears infinitely many times in Pascal's triangle is 1.

Contents 1 Preliminary computations 2 Results and questions 2.1 Do any numbers appear exactly five times in Pascal's triangle? 2.2 Singmaster's conjecture 3 See also 4 References 5 External links

[edit] Preliminary computations

Computation tells us that

• 2 appears just once; all larger positive integers appear more than once;

• 3, 4, 5 each appear 2 times;

• 6 appears 3 times;

Many numbers appear 4 times.

Each of the following appears 6 times:

<math>{120 \choose 1} = {16 \choose 2} = {10 \choose 3}</math>

<math>{210 \choose 1} = {21 \choose 2} = {10 \choose 4}</math>

<math>{1540 \choose 1} = {56 \choose 2} = {22 \choose 3}</math>

<math>{7140 \choose 1} = {239 \choose 2} = {36 \choose 3}</math>

<math>{11628 \choose 1} = {153 \choose 2} = {19 \choose 5}</math>

<math>{24310 \choose 1} = {221 \choose 2} = {17 \choose 8}</math>

The smallest number to appear 8 times is 3003:

<math>{3003 \choose 1} = {78 \choose 2} = {15 \choose 5} = {14 \choose 6}</math>

[edit] Results and questions

David Singmaster (1971) (see References below) defined N(a) to be the multiplicity of the number a within Pascal's triangle. He proved that

<math>N(a) = O(\log a)\,</math>

(see big O notation). Abbot, Erdős, and Hanson (see References) refined the estimate.

Neither of those papers said whether any integer appears exactly five times or exactly seven times.

Singmaster (1975) showed that the diophantine equation

<math>{n+1 \choose k+1} = {n \choose k+2},</math>

has infinitely many solutions for the two variables n, k. It follows that there are infinitely many entries of multiplicity at least 6. The solutions are given by

<math>n = F_{2i+2} F_{2i+3} - 1,\,</math>

<math>k = F_{2i} F_{2i+3} - 1,\,</math>

where F_n is the nth Fibonacci number (indexed according to the convention that F₁ = F₂ = 1).

[edit] Do any numbers appear exactly five times in Pascal's triangle?

It would appear from a related entry, (sequence A003015 in OEIS) in the Online Encyclopedia of Integer Sequences, that no one knows whether the equation N(a) = 5 can be solved for a.

[edit] Singmaster's conjecture

Singmaster (1971) conjectured that there is a finite upper bound on the multiplicities (in the big O notation, N(a) = O(1)). He stated that Erdős concurred and said it could be difficult to prove.

[edit] See also

• Pascal's triangle
• binomial coefficient

[edit] References

• Singmaster, David, "How Often Does an Integer Occur as a Binomial Coefficient?", American Mathematical Monthly, volume 78, number 4, April 1971, pages 385—386.
• Singmaster, David, "Repeated Binomial Coefficients and Fibonacci numbers", Fibonacci Quarterly, volume 13, number 4, pages 296—298, 1975.
• Abbott, H.L., Erdős, P., and Hanson, D., "On the Number of Times an Integer Occurs as a Binomial Coefficient", American Mathematical Monthly, volume 81, number 3, March 1974, pages 256—261.

[edit] External links

• (sequence A003016 in OEIS) (OEIS = Online Encyclopedia of Integer Sequences)