# Project Euler 420
# F(10^7): 2x2 matrices with two distinct positive integer square roots.
import euler.nt { gcd, isqrt }
extern {
function calloc(n: i64, size: i64) -> ptr<void>
function free(p: ptr<void>) -> void
}
function main() -> i32 {
let N: i64 = 10000000
let Umax: i64 = isqrt(N - 2)
let Kmax: i64 = 2 * Umax
let M: i64 = (Kmax * Kmax) / 4
let d: ptr<i32> = calloc(M + 1, 4)
let spf: ptr<i32> = calloc(M + 1, 4)
let expn: ptr<i8> = calloc(M + 1, 1)
let primes: ptr<i32> = calloc(M / 5 + 10, 4)
if d == null || spf == null || expn == null || primes == null { return 1 }
d[1] = 1
let mut pc: i64 = 0
let mut i: i64 = 2
while i <= M {
if spf[i] == 0 {
spf[i] = i as i32
primes[pc] = i as i32
pc = pc + 1
expn[i] = 1
d[i] = 2
}
let si: i64 = spf[i] as i64
let di: i64 = d[i] as i64
let ei: i64 = expn[i] as i64
let mut pi: i64 = 0
while pi < pc {
let p: i64 = primes[pi] as i64
let ip: i64 = i * p
if ip > M { break }
spf[ip] = p as i32
if p == si {
expn[ip] = (ei + 1) as i8
d[ip] = (di / (ei + 1) * (ei + 2)) as i32
break
} else {
expn[ip] = 1
d[ip] = (di * 2) as i32
}
pi = pi + 1
}
i = i + 1
}
# prefix[K] stored flat: for each K, K entries. Total sum_{K=2}^{Kmax} K ~ Kmax^2/2
let pref_size: i64 = (Kmax + 1) * (Kmax + 1)
let prefix: ptr<i32> = calloc(pref_size, 4)
if prefix == null { return 1 }
let mut K: i64 = 2
while K <= Kmax {
let base: i64 = K * (Kmax + 1)
let mut s: i64 = 0
let mut a: i64 = 1
while a < K {
s = s + (d[a * (K - a)] as i64)
prefix[base + a] = s as i32
a = a + 1
}
K = K + 1
}
let mut total: i64 = 0
let mut u: i64 = 2
while u <= Umax {
let mut vmax: i64 = isqrt(N - 1 - u * u)
if vmax >= u { vmax = u - 1 }
let mut v: i64 = 1
while v <= vmax {
let g: i64 = gcd(u, v)
let uu: i64 = u / g
let vv: i64 = v / g
let mut delta: i64 = 1
if (uu % 2 == 1) && (vv % 2 == 1) { delta = 2 }
K = g * delta
if K > 1 {
let s: i64 = u + v
let mut low: i64 = (v * K) / s + 1
let mut high: i64 = (u * K - 1) / s
if low < 1 { low = 1 }
if high > K - 1 { high = K - 1 }
if low <= high {
let base: i64 = K * (Kmax + 1)
total = total + (prefix[base + high] as i64) - (prefix[base + low - 1] as i64)
}
}
v = v + 1
}
u = u + 1
}
printf("%lld\n", total)
free(prefix)
free(primes)
free(expn)
free(spf)
free(d)
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 gcd_i64_i64(int64_t a0, int64_t b0);
int64_t lcm_i64_i64(int64_t a, int64_t b);
int64_t isqrt_i64(int64_t n);
int64_t mulmod_i64_i64_i64(int64_t a0, int64_t b0, int64_t mod);
int64_t mod_pow_i64_i64_i64(int64_t base, int64_t exp, int64_t mod);
bool is_prime_i64(int64_t n);
int32_t main(void);
int64_t gcd_i64_i64(int64_t a0, int64_t b0) {
int64_t a = a0;
int64_t b = b0;
while (b != 0) {
int64_t t = FLOW_CHECKED_MOD((a), (b));
a = b;
b = t;
}
return a;
}
int64_t lcm_i64_i64(int64_t a, int64_t b) {
if ((a == 0 || b == 0)) {
return 0;
}
return (FLOW_CHECKED_DIV((a), (gcd_i64_i64(a, b))) * b);
}
int64_t isqrt_i64(int64_t n) {
if (n < 2) {
return n;
}
int64_t x = n;
int64_t y = FLOW_CHECKED_DIV(((x + 1)), (2));
while (y < x) {
x = y;
y = FLOW_CHECKED_DIV(((x + FLOW_CHECKED_DIV((n), (x)))), (2));
}
return x;
}
int64_t mulmod_i64_i64_i64(int64_t a0, int64_t b0, int64_t mod) {
int64_t a = FLOW_CHECKED_MOD((a0), (mod));
int64_t b = FLOW_CHECKED_MOD((b0), (mod));
int64_t result = 0;
while (b > 0) {
if (FLOW_CHECKED_MOD((b), (2)) == 1) {
result = FLOW_CHECKED_MOD(((result + a)), (mod));
}
a = FLOW_CHECKED_MOD(((a * 2)), (mod));
b = FLOW_CHECKED_DIV((b), (2));
}
return result;
}
int64_t mod_pow_i64_i64_i64(int64_t base, int64_t exp, int64_t mod) {
if (mod == 1) {
return 0;
}
int64_t result = 1;
int64_t b = FLOW_CHECKED_MOD((base), (mod));
int64_t e = exp;
while (e > 0) {
if (FLOW_CHECKED_MOD((e), (2)) == 1) {
result = mulmod_i64_i64_i64(result, b, mod);
}
b = mulmod_i64_i64_i64(b, b, mod);
e = FLOW_CHECKED_DIV((e), (2));
}
return result;
}
bool is_prime_i64(int64_t n) {
if (n < 2) {
return 0;
}
if (n < 4) {
return 1;
}
if ((FLOW_CHECKED_MOD((n), (2)) == 0 || FLOW_CHECKED_MOD((n), (3)) == 0)) {
return 0;
}
int64_t i = 5;
while ((i * i) <= n) {
if ((FLOW_CHECKED_MOD((n), (i)) == 0 || FLOW_CHECKED_MOD((n), ((i + 2))) == 0)) {
return 0;
}
i = (i + 6);
}
return 1;
}
int32_t main(void) {
int64_t N = 10000000;
int64_t Umax = isqrt_i64((N - 2));
int64_t Kmax = (2 * Umax);
int64_t M = FLOW_CHECKED_DIV(((Kmax * Kmax)), (4));
int32_t* d = (int32_t*)(calloc((M + 1), 4));
int32_t* spf = (int32_t*)(calloc((M + 1), 4));
int8_t* expn = (int8_t*)(calloc((M + 1), 1));
int32_t* primes = (int32_t*)(calloc((FLOW_CHECKED_DIV((M), (5)) + 10), 4));
if ((((d == NULL || spf == NULL) || expn == NULL) || primes == NULL)) {
return 1;
}
d[1] = 1;
int64_t pc = 0;
int64_t i = 2;
while (i <= M) {
if (spf[i] == 0) {
spf[i] = ((int32_t)(i));
primes[pc] = ((int32_t)(i));
pc = (pc + 1);
expn[i] = 1;
d[i] = 2;
}
int64_t si = ((int64_t)(spf[i]));
int64_t di = ((int64_t)(d[i]));
int64_t ei = ((int64_t)(expn[i]));
int64_t pi = 0;
while (pi < pc) {
int64_t p = ((int64_t)(primes[pi]));
int64_t ip = (i * p);
if (ip > M) {
break;
}
spf[ip] = ((int32_t)(p));
if (p == si) {
expn[ip] = ((int8_t)((ei + 1)));
d[ip] = ((int32_t)((FLOW_CHECKED_DIV((di), ((ei + 1))) * (ei + 2))));
break;
} else {
expn[ip] = 1;
d[ip] = ((int32_t)((di * 2)));
}
pi = (pi + 1);
}
i = (i + 1);
}
int64_t pref_size = ((Kmax + 1) * (Kmax + 1));
int32_t* prefix = (int32_t*)(calloc(pref_size, 4));
if (prefix == NULL) {
return 1;
}
int64_t K = 2;
while (K <= Kmax) {
int64_t base = (K * (Kmax + 1));
int64_t s = 0;
int64_t a = 1;
while (a < K) {
s = (s + ((int64_t)(d[(a * (K - a))])));
prefix[(base + a)] = ((int32_t)(s));
a = (a + 1);
}
K = (K + 1);
}
int64_t total = 0;
int64_t u = 2;
while (u <= Umax) {
int64_t vmax = isqrt_i64(((N - 1) - (u * u)));
if (vmax >= u) {
vmax = (u - 1);
}
int64_t v = 1;
while (v <= vmax) {
int64_t g = gcd_i64_i64(u, v);
int64_t uu = FLOW_CHECKED_DIV((u), (g));
int64_t vv = FLOW_CHECKED_DIV((v), (g));
int64_t delta = 1;
if ((FLOW_CHECKED_MOD((uu), (2)) == 1 && FLOW_CHECKED_MOD((vv), (2)) == 1)) {
delta = 2;
}
K = (g * delta);
if (K > 1) {
int64_t s = (u + v);
int64_t low = (FLOW_CHECKED_DIV(((v * K)), (s)) + 1);
int64_t high = FLOW_CHECKED_DIV((((u * K) - 1)), (s));
if (low < 1) {
low = 1;
}
if (high > (K - 1)) {
high = (K - 1);
}
if (low <= high) {
int64_t base = (K * (Kmax + 1));
total = ((total + ((int64_t)(prefix[(base + high)]))) - ((int64_t)(prefix[((base + low) - 1)])));
}
}
v = (v + 1);
}
u = (u + 1);
}
printf("%lld\n", total);
free(prefix);
free(primes);
free(expn);
free(spf);
free(d);
return 0;
}