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Singmaster’s conjecture, named after David Singmaster, on how many times different numbers appear in Pascal’s triangle. It states that, except for one, there is a maximum number that any number will appear.
Obviously, the number 1 will appear an infinite number of times, since every row starts and ends with 1. However, the second diagonal in Pascal’s triangle contains the counting numbers 2,3,4,5… , as well as the number you choose, a sequence that is the second and second last number. After this sequence, all numbers except 1 will be greater than the number you selected. Therefore, we can conclude that every number except 1 appears several times in Pascal’s triangle.
This is what the Singmaster asked: is there an infinite number of iterations that cannot be exceeded anything number? For example, can we say that whatever number you choose, no matter how big, it will not appear more than 20 times in Pascal’s triangle? Here are some examples of how often different numbers appear in Pascal’s triangle:
1 appears infinitely many times
2 is the only number that appears only once
3,4,5,7,8 all appear only twice
6 appears three times
10 appears four times
120, 210, 1540 appear six times each
3003 appears eight times
These examples show that there are surprisingly few repetitions of numbers in Pascal’s triangle, with most numbers appearing only twice (as the second and second digits in a row). In addition, no other number has been found to appear more than 6 times except 3003, and no number has been found to appear five or seven times, but no one knows for sure whether such numbers exist.
However, before you turn away in disgust from useless and lazy mathematicians, Singmaster noticed something interesting about numbers that appear six or more times – he proved that there are infinitely many of them. In fact, he found a formula that always gives you numbers like this:
n = F(2i+2)*F(2i+3)-1
k = F(2i)F(2i+3) +1
Here F(I) represents the Fibonacci number of I (numbers from the sequence 1,1,2,3,5,8,13,21,34,55… where each number is the sum of the previous two). Once you have calculated n and k, to get the actual number that appears six or more times, you must calculate n = n!/(r!(nr)!). Therefore, if we choose I = 1 as an example, we get
n = F(2*1+2)F(2*1+3) -1
= F(4)F(5) – 1
= 3*5 -1 (The fourth and fifth Fibonacci numbers are 3 and 5.)
and k = F(2*1)F(2*1+3) +1
= F(2)F(5) + 1
= 1*5 +1
Finally, we calculate 14 select 6:
So the first number from the Singmaster formula that has 6 or more appearances in Pascal’s triangle is 3003. Two things are worth noting here: First, the number formula does not give you exactly 6 repetitions, but 6 or more. So even though 3003 actually appears 8 times, the formula is still valid. Second, the Singmaster formula does not work everybody number with six or more digits. Recall from our previous examples that 120, 210 and 1540 are all smaller examples of 3003 with six or more appearances in Pascal’s triangle. However, you can replace the “I” in the formula with any number you like, so the Singmaster formula still shows an infinite number of digits appearing 6 or more times.
By the way, the next number you get from the Singmaster formula (using I = 2) is 61218182743304701891431482520.
I find it very interesting how something as simple as Pascal’s triangle, which after all, is just a series of very simple additions, can lead to such mysteries. It is humbling to think that such simple questions that are so easy to put together, the answers to which continue to confuse us and the answers to which are still elusive.
If you would like to learn more about Pascal’s triangle and its amazing patterns, properties and mysteries, including how it relates to the Fibonacci numbers (which here seemed to just come out of thin air), then please Feel free to visit my website by following the link. in the resource box below.
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