Way

There is always a WAY.

.

(defun sum-of-squares(x y &key (a 1) (b 1))
  (+ (* a x x) (* b y y)))

(deftype candidate()
  '(simple-array boolean (*)))

(defmacro get-sieve-function(a b mod-results)
  `(lambda(x y)
     (let* ((n (sum-of-squares x y :a ,a :b ,b))
        (r (mod n 12)))
       (when (and (<= n limit)
          (not (null
            (member r ,mod-results))))
     (setf (aref candidates n)
           (not (aref candidates n)))))))

(defun get-atkin-prime-candidates-map(limit)
  (let* ((lmt (isqrt limit))
         (candidates (make-array
              (1+ limit)
              :initial-element nil))
         (stage1 (get-sieve-function 4 1 '(1 5)))
         (stage2 (get-sieve-function 3 1 '(7)))
         (stage3 (get-sieve-function 3 -1 '(11))))
    (declare (type candidate candidates))
    (declare (optimize (speed 3)))
    (loop for x from 1 to lmt do
      (loop for y from 1 to lmt do
    (progn
      (funcall stage1 x y)
      (funcall stage2 x y)
      (when (> x y)
        (funcall stage3 x y)))))
    candidates))

(defun atkin-sieve-map(candidates)
  (let ((len (length candidates)))
    (loop for i from 1 to (1- len)
      when (aref candidates i)
        do (loop for j from 1
             for n = (* j i i)
             while (< n len)
             do (setf (aref candidates n) nil))))
  (setf (aref candidates 2) T)
  (setf (aref candidates 3) T)
  candidates)

(defun eratosthene-sieve-map(limit)
  (let ((candidates
      (make-array
       (1+ limit) :initial-element T)))
    (declare (type candidate candidates))
    (declare (optimize (speed 3)))
    (progn
      (setf (aref candidates 0) nil)
      (setf (aref candidates 1) nil))
    (loop for i from 2 to limit
      when (aref candidates i)
        do (loop for j from (+ i i) to limit by i
             when (aref candidates j)
               do (setf (aref candidates j) nil)))
    candidates))

;; This is free software released
;; into the public domain.
;;
;; ykm

(defun get-primes(limit
          &key
            (generator :eratosthene))
  (let ((candidates
      (case generator
        (:atkin (atkin-sieve-map
             (get-atkin-prime-candidates-map
              limit)))
        (:eratosthene (eratosthene-sieve-map
               limit))
        (otherwise
         (error "not valid type
(:atkin :eratosthene)")))))
    (declare (type candidate candidates))
    (loop for i from 0 to (1- (length candidates))
      when (aref candidates i)
        collect i)))

(format t "soe: ~d
" (time (length (get-primes 12345678))))

(format t "soa: ~d
" (time (length (get-primes 12345678
                :generator :atkin))))

(defmacro log10 (x)
  `(/ (log ,x) (log 10)))

.

— Me@2022.09.06 03:24:36 PM

.

.

2022.09.06 Tuesday (c) All rights reserved by ACHK

Homogenous function

A function f is homogenous of degree n if and only if f(ax) = a^n f(x).

— 1.8 Conserved Quantities

— Structure and Interpretation of Classical Mechanics

.

\displaystyle{\begin{aligned}     D_a f(ax)     &= \frac{d}{da} f(ax) \\     &= \frac{d}{da} a^n f(x) \\     &= f(x) \frac{d}{da} a^n  \\     &= n a^{n-1} f(x) \\     \end{aligned}}

.

\displaystyle{\begin{aligned}     D_a f(ax)     &= \frac{d}{da} f(ax) \\     &= \frac{d(ax)}{da} \frac{d}{d(ax)} f(ax) \\     &= x \frac{d}{d(ax)} f(ax) \\     &= x D_{ax} f(ax) \\     \end{aligned}}

.

\displaystyle{\begin{aligned}     x D_{ax} f(ax) &= n a^{n-1} f(x) \\    x D_{x} f(x) &= n f(x) \\    \end{aligned}}

— Me@2022.09.05 06:28:00 PM

.

.

2022.09.06 Tuesday (c) All rights reserved by ACHK