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All centered square numbers and their divisors have a remainder of 1 when divided by 4. Hence all centered square numbers and their divisors end with digit 1 or 5 in base 6, 8, and 12. Every centered square number except 1 is the hypotenuse of a Pythagorean triple (3-4-5, 5-12-13, 7-24-25, ...). This is exactly the sequence of Pythagorean ...
if the last digit of a number is 4 or 6, its square ends in an odd digit followed by a 6; and; if the last digit of a number is 5, its square ends in 25. In base 12, a square number can end only with square digits (like in base 12, a prime number can end only with prime digits or 1), that is, 0, 1, 4 or 9, as follows:
[6] When Marion Walter, who was also part of the project in the 1960s spoke to Prenowitz in 1996, he said that he considered the allocation of one color to all blocks of a particular shape, much like Cuisenaire rods, which may have given him the idea, to be one of the innovative features of the blocks. Also important in his choice was that ...
Whereas a prime number p cannot be a polygonal number (except the trivial case, i.e. each p is the second p-gonal number), many centered polygonal numbers are primes. In fact, if k ≥ 3, k ≠ 8, k ≠ 9, then there are infinitely many centered k -gonal numbers which are primes (assuming the Bunyakovsky conjecture ).
The square of an integer may also be called a square number or a perfect square. In algebra, the operation of squaring is often generalized to polynomials, other expressions, or values in systems of mathematical values other than the numbers. For instance, the square of the linear polynomial x + 1 is the quadratic polynomial (x + 1) 2 = x 2 ...
As the square of a Catalan number, it counts the number of walks of length 8 in the positive quadrant of the integer grid that start and end at the origin, moving diagonally at each step. [1] It is part of a sequence of square numbers beginning 0, 1, 4, 25, 196, ... in which each number is the smallest square that differs from the previous ...