Symmetric Diophantine Equation. S(N) = sum (x+y+z) over primitive solutions of 15(x^2+y^2+z^2) = 34(xy+yz+zx) with 1 <= x <= y <= z <= N and gcd(x,y,z)=1. Coprime-pair parameterisation with SPF sieve.
# Project Euler 785
# Symmetric Diophantine Equation.
# S(N) = sum (x+y+z) over primitive solutions of
# 15(x^2+y^2+z^2) = 34(xy+yz+zx)
# with 1 <= x <= y <= z <= N and gcd(x,y,z)=1.
# Coprime-pair parameterisation with SPF sieve.
extern {
function calloc(n: i64, size: i64) -> ptr<void>
function free(p: ptr<void>) -> void
function sqrt(x: f64) -> f64
}
function isqrt_i64(x: i64) -> i64 {
if x <= 0 { return 0 }
let mut r: i64 = sqrt(x as f64) as i64
while r * r > x {
r = r - 1
}
while (r + 1) * (r + 1) <= x {
r = r + 1
}
return r
}
function main() -> i32 {
let N: i64 = 1000000000
let max_a: i64 = isqrt_i64(N / 3)
if max_a < 1 {
printf("%lld\n", 0 as i64)
return 0
}
# SPF sieve
let spf: ptr<i32> = calloc(max_a + 1, 4) as ptr<i32>
for i in 0..(max_a + 1) {
spf[i] = i as i32
}
let mut i: i64 = 2
while i * i <= max_a {
if spf[i] == i as i32 {
let mut j: i64 = i * i
while j <= max_a {
if spf[j] == j as i32 {
spf[j] = i as i32
}
j = j + i
}
}
i = i + 1
}
let N3: i64 = 12 * N
let N5: i64 = 5 * N
let mut total: i64 = 0
let pf: ptr<i32> = calloc(64, 4) as ptr<i32>
for a in 1..(max_a + 1) {
let aa: i64 = a * a
# distinct prime factors of a
let mut pf_n: i64 = 0
let mut x: i64 = a
while x > 1 {
let p: i64 = spf[x] as i64
pf[pf_n] = p as i32
pf_n = pf_n + 1
while x % p == 0 {
x = x / p
}
}
# bmin from B = 5a^2 - 2ab <= N ==> b >= (5a^2 - N)/(2a)
let mut bmin: i64
if 5 * aa <= N {
bmin = 1
} else {
let num: i64 = 5 * aa - N
bmin = (num + (2 * a - 1)) / (2 * a)
if bmin < 1 { bmin = 1 }
}
# C > 0 requires b < 3a/5
let bmax_pos: i64 = (3 * a - 1) / 5
if bmax_pos < bmin { continue }
# A = 2ab + 3b^2 <= N -> b <= floor((-2a + sqrt(4a^2+12N))/6)
let disc1: i64 = 4 * aa + N3
let bmax1: i64 = (-2 * a + isqrt_i64(disc1)) / 6
# C = 3a^2 - 8ab + 5b^2 <= N
let disc3: i64 = aa + N5
let sdisc3: i64 = isqrt_i64(disc3)
let bmax3: i64 = (4 * a + sdisc3) / 5
let bmin3: i64 = (4 * a - sdisc3 + 4) / 5
if bmin3 > bmin { bmin = bmin3 }
let mut bmax: i64 = bmax_pos
if bmax1 < bmax { bmax = bmax1 }
if bmax3 < bmax { bmax = bmax3 }
if bmax >= a { bmax = a - 1 }
if bmax < bmin { continue }
let mut b: i64 = bmin
while b <= bmax {
# coprimality: b not divisible by any prime factor of a
let mut ok: i64 = 1
let mut k: i64 = 0
while k < pf_n {
if b % (pf[k] as i64) == 0 {
ok = 0
break
}
k = k + 1
}
if ok == 0 {
b = b + 1
continue
}
let ab: i64 = a * b
let bb: i64 = b * b
let A: i64 = 2 * ab + 3 * bb
let B: i64 = 5 * aa - 2 * ab
let C: i64 = 3 * aa - 8 * ab + 5 * bb
if C <= 0 {
b = b + 1
continue
}
if A > N || B > N || C > N {
b = b + 1
continue
}
# non-primitive iff all divisible by 19
if A % 19 == 0 && B % 19 == 0 {
b = b + 1
continue
}
total = total + 8 * (aa - ab + bb)
b = b + 1
}
}
free(spf)
free(pf)
printf("%lld\n", total)
return 0
}
Generated C
#include <stdint.h>
#include <stdbool.h>
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
/* Flow runtime helpers */
typedef struct flow_temp_node { struct flow_temp_node* next; } flow_temp_node;
static flow_temp_node* flow_temp_head = NULL;
static int flow_temp_atexit_set = 0;
__attribute__((unused)) static void flow_temp_free_all(void) {
while (flow_temp_head) {
flow_temp_node* n = flow_temp_head;
flow_temp_head = n->next;
free(n);
}
}
__attribute__((unused)) static void* flow_temp_alloc(size_t nbytes) {
flow_temp_node* node = (flow_temp_node*)malloc(sizeof(flow_temp_node) + nbytes);
if (!node) return NULL;
node->next = flow_temp_head;
flow_temp_head = node;
if (!flow_temp_atexit_set) {
flow_temp_atexit_set = 1;
atexit(flow_temp_free_all);
}
return (void*)(node + 1);
}
#ifndef FLOW_DIAG
#define FLOW_DIAG(msg) fprintf(stderr, "%s", (msg))
#endif
#ifndef FLOW_LOG
#define FLOW_LOG(fmt, ...) printf(fmt, __VA_ARGS__)
#endif
#ifndef FLOW_LOG_EMPTY
#define FLOW_LOG_EMPTY(fmt) printf(fmt)
#endif
static char* flow_strcat(const char* a, const char* b) {
size_t la = strlen(a ? a : ""), lb = strlen(b ? b : "");
char* r = (char*)flow_temp_alloc(la + lb + 1);
if (!r) return NULL;
if (la) memcpy(r, a, la);
if (lb) memcpy(r + la, b, lb);
r[la + lb] = '\0';
return r;
}
#define __flow_in_arr(arr, val) __extension__ ({ \
int _found = 0; \
size_t _n = sizeof(arr)/sizeof((arr)[0]); \
for (size_t _i = 0; _i < _n; _i++) { \
if ((arr)[_i] == (val)) { _found = 1; break; } \
} _found; })
/* Unified fault handler (MISRA #279) — override with -DFLOW_FAULT_HANDLER=fn */
#ifndef FLOW_FAULT_HANDLER
__attribute__((unused)) static inline void flow_fault_handler(const char* msg) {
fprintf(stderr, "flow: %s\n", msg ? msg : "fault");
abort();
#if defined(__GNUC__) || defined(__clang__)
__builtin_unreachable();
#endif
}
#else
#define flow_fault_handler FLOW_FAULT_HANDLER
#endif
#define flow_div_by_zero_handler() flow_fault_handler("division by zero")
#define flow_shift_ub_handler() flow_fault_handler("invalid shift (amount out of range or left-shift of negative)")
#ifndef FLOW_CHECKED_DIV
#define FLOW_CHECKED_DIV(L, R) (((R) != 0) ? ((L) / (R)) : (flow_div_by_zero_handler(), (L) * 0))
#endif
#ifndef FLOW_CHECKED_MOD
#define FLOW_CHECKED_MOD(L, R) (((R) != 0) ? ((L) % (R)) : (flow_div_by_zero_handler(), (L) * 0))
#endif
#ifndef FLOW_CHECKED_SHL
#define FLOW_CHECKED_SHL(L, R) ((((R) >= 0) && ((unsigned long long)(R) < (sizeof(L) * 8ull)) && ((L) >= 0)) ? ((L) << (R)) : (flow_shift_ub_handler(), (L) * 0))
#endif
#ifndef FLOW_CHECKED_SHR
#define FLOW_CHECKED_SHR(L, R) ((((R) >= 0) && ((unsigned long long)(R) < (sizeof(L) * 8ull))) ? ((L) >> (R)) : (flow_shift_ub_handler(), (L) * 0))
#endif
#include <math.h>
void* _ui_state = NULL;
static inline float i32_to_f32(int32_t v) { return (float)v; }
/* Host stub for @gpu kernels (device codegen replaces this). */
static inline int32_t gpu_thread_id(void) { return 0; }
int64_t isqrt_i64_i64(int64_t x);
int32_t main(void);
int64_t isqrt_i64_i64(int64_t x) {
if (x <= 0) {
return 0;
}
int64_t r = ((int64_t)(sqrt(((double)(x)))));
while ((r * r) > x) {
r = (r - 1);
}
while (((r + 1) * (r + 1)) <= x) {
r = (r + 1);
}
return r;
}
int32_t main(void) {
int64_t N = 1000000000;
int64_t max_a = isqrt_i64_i64(FLOW_CHECKED_DIV((N), (3)));
if (max_a < 1) {
printf("%lld\n", ((int64_t)(0)));
return 0;
}
int32_t* spf = (int32_t*)(((int32_t*)(calloc((max_a + 1), 4))));
int32_t __flow_step_1 = 1;
for (int32_t i = 0; (0 <= (max_a + 1)) ? i < (max_a + 1) : i > (max_a + 1); i += (0 <= (max_a + 1)) ? 1 : -1) {
spf[i] = ((int32_t)(i));
}
int64_t i = 2;
while ((i * i) <= max_a) {
if (spf[i] == ((int32_t)(i))) {
int64_t j = (i * i);
while (j <= max_a) {
if (spf[j] == ((int32_t)(j))) {
spf[j] = ((int32_t)(i));
}
j = (j + i);
}
}
i = (i + 1);
}
int64_t N3 = (12 * N);
int64_t N5 = (5 * N);
int64_t total = 0;
int32_t* pf = (int32_t*)(((int32_t*)(calloc(64, 4))));
int32_t __flow_step_2 = 1;
for (int32_t a = 1; (1 <= (max_a + 1)) ? a < (max_a + 1) : a > (max_a + 1); a += (1 <= (max_a + 1)) ? 1 : -1) {
int64_t aa = (a * a);
int64_t pf_n = 0;
int64_t x = a;
while (x > 1) {
int64_t p = ((int64_t)(spf[x]));
pf[pf_n] = ((int32_t)(p));
pf_n = (pf_n + 1);
while (FLOW_CHECKED_MOD((x), (p)) == 0) {
x = FLOW_CHECKED_DIV((x), (p));
}
}
int64_t bmin;
if ((5 * aa) <= N) {
bmin = 1;
} else {
int64_t num = ((5 * aa) - N);
bmin = FLOW_CHECKED_DIV(((num + ((2 * a) - 1))), ((2 * a)));
if (bmin < 1) {
bmin = 1;
}
}
int64_t bmax_pos = FLOW_CHECKED_DIV((((3 * a) - 1)), (5));
if (bmax_pos < bmin) {
continue;
}
int64_t disc1 = ((4 * aa) + N3);
int64_t bmax1 = FLOW_CHECKED_DIV(((((-2) * a) + isqrt_i64_i64(disc1))), (6));
int64_t disc3 = (aa + N5);
int64_t sdisc3 = isqrt_i64_i64(disc3);
int64_t bmax3 = FLOW_CHECKED_DIV((((4 * a) + sdisc3)), (5));
int64_t bmin3 = FLOW_CHECKED_DIV(((((4 * a) - sdisc3) + 4)), (5));
if (bmin3 > bmin) {
bmin = bmin3;
}
int64_t bmax = bmax_pos;
if (bmax1 < bmax) {
bmax = bmax1;
}
if (bmax3 < bmax) {
bmax = bmax3;
}
if (bmax >= a) {
bmax = (a - 1);
}
if (bmax < bmin) {
continue;
}
int64_t b = bmin;
while (b <= bmax) {
int64_t ok = 1;
int64_t k = 0;
while (k < pf_n) {
if (FLOW_CHECKED_MOD((b), (((int64_t)(pf[k])))) == 0) {
ok = 0;
break;
}
k = (k + 1);
}
if (ok == 0) {
b = (b + 1);
continue;
}
int64_t ab = (a * b);
int64_t bb = (b * b);
int64_t A = ((2 * ab) + (3 * bb));
int64_t B = ((5 * aa) - (2 * ab));
int64_t C = (((3 * aa) - (8 * ab)) + (5 * bb));
if (C <= 0) {
b = (b + 1);
continue;
}
if (((A > N || B > N) || C > N)) {
b = (b + 1);
continue;
}
if ((FLOW_CHECKED_MOD((A), (19)) == 0 && FLOW_CHECKED_MOD((B), (19)) == 0)) {
b = (b + 1);
continue;
}
total = (total + (8 * ((aa - ab) + bb)));
b = (b + 1);
}
}
free(spf);
free(pf);
printf("%lld\n", total);
return 0;
}