Text: 1. All integers are divided into two classes: odd and even. 2. All even integers are divided into two classes: 2-par, (4n-2) and 4-par, (4n) integers. 3. All odd integers are divided into two classes: 3-par (4n-1), and 5-par, (4n+1). 4. Only 4-par and 5-par numbers may be squares of integers. 2-par and 3-par integers may not, but all four may be rectangles. 5. Both odd and even integers further divides each of its classes into pairs of classes on a binary basis, around 8, 16, 32, 64, etc. 6. The base, C, of a triangle, and the line joining foci of an ellipse must be a 4-par integer. (Doubling changes this to 2-par.) This 4-par integer is the foundation number in every mathematical case in Quantum Arithmetic around which all the integers of a detail will build. And in the vein of Pseudaria, this 4-par integer maintains its group of integers in a strict orientation and polarity when they build upon it. This base is always divisible by four in its prime useages, (but divisible by 2 after a doubling takes place.) 7. The altitude F, of a triangle, is always divisible by three, or by some higher odd integer. When not divisible by three, then the base is divisible by three as well as by four. (i.e., divisible by 12). The altitude of a triangle will form a gnomon which is the difference of two squares, and a rectangle, in as many as it has divisors less than the square root of the integer representing it. 8. The hypotenuse, G, of a triangle, must be divisible by 5, or be some higher integer which is 5-par. It cannot be divisible by 3 or 4. If G is not divisible by 5 then the altitude, F, must be divisible by 5. If the hypotenuse or the altitude are not divisible by 5 then the base must be so divisible in addition to its other duties. The hypotenuse must be the sum of two squares, (D and E), and must be the mean of two squares, (B and A). 9. The mean of the altitude, F, and the hypotenuse, G, must be a square number, D, and must differ from them by another square, E, which is of an opposite parity. Of these two squares, one must be 4-par, and the other must be 5-par. 10. The value of the hypotenuse, G, as the mean of two sqares, A and B, will differ from them by a 4-par number, C, and both squares must be odd and 5-par integers. 11. Every operation with a group of connected integers will have a system of four roots, b, e, d, and a, underlying the operation. One of these roots, d or e, must be even, and the other three must be odd. (Doubling changes the rule.) 12. One of the four roots must be divisible by three. 13. One of the four roots must be divisible by five, or the sum of the squares of the two center ones, d and e, must be divisble by 5. (vis. must be 5-pent or 5n in character.)

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