In this paper, we investigate arithmetic progressions in the polygonal numbers with a fixed number of sides. We first show that four-term arithmetic progressions cannot exist. We then describe explicitly how to find all three-term arithmetic progressions. Finally, we show that not only are there infinitely many three-term arithmetic progressions, but that there are infinitely many three-term ar...

This paper presents complete processor hardware with three arithmetic units. The first arithmetic unit can perform 32-bit integer arithmetic operations. The second unit can perform arithmetic operations such as addition, subtraction, multiplication, division, and square root on 32-bit floating point numbers. The third unit can perform arithmetic operations such as addition, subtraction, multipl...

Let p > 3 be a prime. We derive the following new congru-ences: p−1 n=0 (2n + 1)A n ≡ p (mod p 4) and p−1 n=0

We investigate some arithmetic properties of the q-Fibonacci numbers and the q-Pell numbers.

I here recall Ryle's analysis of Heterologicality, but broaden the discussion to comparable analyses not only of Heterologicality but also other puzzles about self-reference. Such matters have a crucial bearing on the debate between representational and non-representational theories of mind, as will be explained. 1 [1] The most remarkable thing about the so-called Heterologicality Paradox is th...

In a previous paper we investigated Givens transformations applied to quaternion valued matrices. Since arithmetic operations with quaternions are very costly it is desirable to reduce the number of arithmetic operations with quaternions. We show that the Fast Givens transformation, known for the real case, can also be defined for quaternion valued matrices, and we apply this technique to the r...

In a previous paper we investigated Givens transformations applied to quaternion valued matrices. Since arithmetic operations with quaternions are very costly it is desirable to reduce the number of arithmetic operations with quaternions. We show that the Fast Givens transformation, known for the real case, can also be defined for quaternion valued matrices, and we apply this technique to the r...