Last week we asked about a formula that would allow us to find the nth tetrahedral number as a function of n without having to add the first n triangular numbers; Here it is (can you prove it?):

Tn = n(n + 1)(n + 2)/6

In the case of n = 22:

22 x 23 x 24/6 = 2024

Therefore, the formula confirms that 2024 is the twenty-second tetrahedral number.

As for last week's second question, there are four numbers that are both tetrahedral and triangular. The first two are easy to find and the other two are not so easy: 10, 120, 1540 and 7140 (is it a coincidence that they all end in 0? ?), which are the third, eighth, twentieth and thirty-fourth tetrahedral numbers, respectively (as well as the 4th, 15th, 55th and 119th triangular numbers).

And as for the third question, the most difficult (not to say impossible at the level of recreational mathematics): there are only three tetrahedral numbers that are perfect squares, as AJ Meyl demonstrated in 1878. The first two are trivial: T1 = 1 and T2 = 4, but the third is hardly achievable: T48 = 1402 = 19600.

By the way, the only tetrahedral number that is also a square pyramid number is 1, as Dutch mathematician Frits Beukers demonstrated in 1988. Note the contrast between the ease of finding 1 as a random number and the difficulty of proving that it is the only one.

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### Pascal's Triangle, Tartaglia, Jayam…

If we look at the famous Pascal triangle, also known as the Tartaglia triangle, in which in each row the numbers between the side ones are the sum of the two directly above them, we see that the third diagonal is both right and right On the left is the sequence of triangular numbers: 1, 3, 6, 10, 15, 21, 28…, while on the fourth diagonal we have the tetrahedral numbers: 1, 4, 10, 20, 35, 56…

In the West, this fascinating number triangle is known as Pascal's or Tartaglia's triangle, in honor of the French mathematician and the Italian algebraist who studied it in detail; but in reality it was known in the East long before. In the 11th century, the Persian mathematicians Al-Karayí and Omar Khayam analyzed its properties in detail, which is why it is known as the Khayam triangle in Iran and other eastern countries. And the Chinese, like almost everything, have their own predecessors, such as Jia. In China, the number triangle is known as the Yang-Hui triangle.

And if this triangle has many names, it holds many more mathematical treasures. I invite my clever readers to look for some:

How is the Jayam triangle (I prefer to call it in honor of the great Persian poet and mathematician) related to the number e?

Can we find the Fibonacci sequence in it?

Can you use this to determine the primality of a number?

However, as far as I know, and despite the fact that the number π appears where you least expect it, there is no way to connect it to our versatile number triangle. Or when?

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