2024-01-19 13:42:26
Last week we asked ourselves 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 easy to find and the other two not so easy: 10, 120, 1540 and 7140 (is it a coincidence that they all end in 0?), which are, respectively, the third, eighth, twentieth and thirty-fourth tetrahedral number (as well as the 4th, 15th, 55th and 119th triangular number).
And regarding 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.
Incidentally, the only tetrahedral number that is also a square pyramidal number is 1, as the Dutch mathematician Frits Beukers demonstrated in 1988. Note the contrast between the ease of finding 1 as a coincident number and the difficulty of proving that it is the only.
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Pascal’s triangle, Tartaglia, Jayam…
If we look at the famous Pascal’s triangle, also known as Tartaglia’s triangle, in which, in each row, the numbers between the lateral 1’s are the sum of the two just above it, we see that the third diagonal, both along the right and 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 numerical triangle is known as Pascal’s or Tartaglia’s triangle, in honor of the French mathematician and the Italian algebraist who studied it in depth; but in reality it was already known in the East long before. In the 11th century, the Persian mathematicians Al-Karayí and Omar Khayam extensively analyzed its properties, which is why in Iran and other eastern countries it is known as the Khayam triangle. And the Chinese, as in almost everything, have their own precursors, such as Jia In China the numerical triangle is known as the Yang Hui triangle.
And if this triangle has many names, it hides many more mathematical treasures. I invite my sagacious 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 locate the Fibonacci sequence in it?
Can it be used to determine the primality of a number?
However, as far as I know, and despite the fact that the number π appears where it is least expected, there is no way to relate it to our versatile numerical triangle. Or if?
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