It gives me great pleasure NOT to say...
"----- pls wait a few days before posting answers or hints."
_______________________
Where n (>0) is a (positive) integer....
Some forms are obviously squares: n^2, n^4, n^6, .........
Why don't you suggest some forms that are (at least a bit) puzzling.
e.g. -------
n^2 -1, n^2 +1, n^4 + n, n^6 - n, ............
n^3, n^5, n^7 - 1, ............
n(n+1), n(n+1)(n+2), n(n+1)(n+2)(n+3), ............
Some of these forms are NEVER squares,
and others are sometimes squares.
___________________________________
(Sum of 2 squares) Pythagorean triple : (odd) ^ 2 + (odd) ^ 2 = (integer) ^ 2
We've known about Squares ( and Pythagorean triples )
for so long that.... When we learn something
simple and new about them, it's like....
_______ about a favorite grand-father.
On Thursday, July 21, 2022 at 1:26:55 PM UTC-4, henh...@gmail.com wrote:
It gives me great pleasure NOT to say...
"----- pls wait a few days before posting answers or hints."
_______________________
Where n (>0) is a (positive) integer....
Some forms are obviously squares: n^2, n^4, n^6, .........
Why don't you suggest some forms that are (at least a bit) puzzling.
e.g. -------
n^2 -1, n^2 +1, n^4 + n, n^6 - n, ............
n^3, n^5, n^7 - 1, ............
n(n+1), n(n+1)(n+2), n(n+1)(n+2)(n+3), ............
Some of these forms are NEVER squares,
and others are sometimes squares.
___________________________________
(Sum of 2 squares) Pythagorean triple : (odd) ^ 2 + (odd) ^ 2 = (integer) ^ 2
which don't.We've known about Squares ( and Pythagorean triples )The "polygonal numbers" are a set of sequences, one for each integer n. They are defined as x*((n-2)x+(4-n))/2.
for so long that.... When we learn something
simple and new about them, it's like....
_______ about a favorite grand-father.
So for example, the 3-gonal (triangle) numbers are x(x+1)/2; the 4-gonal (square numbers) are x*x, and the 5- (pentagonal) numbers are x(3x-1)/2.
Some triangle numbers are square; which ones is an interesting, and classic, problem that has been connected with Ramanujan.
All square numbers are square.
I don't know the answer for the other -gons: I suspect it's similar to the triangles (an interesting pattern of which ones are square). I can say that any that have solutions will have similar series of solutions, but I don't know if there are any
The squangular numbers:
1, 36, 1225, 41616, 1413721, 48024900, 1631432881, ... (numbers that
are both square and triangular).
.............. are both square [n^2] and
triangular [n(n+1)/2].
Next ten? Is there a formula for them?
In article <9ad9c317-744e-4191-8be6-18749644fd9an@googlegroups.com>, >henh...@gmail.com <henhanna@gmail.com> wrote:
The squangular numbers:
1, 36, 1225, 41616, 1413721, 48024900, 1631432881, ... (numbers that
are both square and triangular).
.............. are both square [n^2] and
triangular [n(n+1)/2].
Next ten? Is there a formula for them?
Strange but true:
(sinh(2*n*asinh(1)))**2 / 8
The Online Encyclopedia of Integer Sequences is your friend:
https://oeis.org/A001110
-- Richard
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