# Project Euler 216
# Count primes of form t(n)=2n^2-1 for 2<=n<=50e6.
import euler.nt { isqrt }
extern {
function calloc(n: i64, size: i64) -> ptr<void>
function free(p: ptr<void>) -> void
}
function modpow(base: i64, exp: i64, mod: i64) -> i64 {
let mut r: i64 = 1
let mut b: i64 = base % mod
let mut e: i64 = exp
while e > 0 {
if e % 2 == 1 { r = (r * b) % mod }
b = (b * b) % mod
e = e / 2
}
return r
}
function tonelli(a0: i64, p: i64) -> i64 {
let a: i64 = a0 % p
if a == 0 { return 0 }
if modpow(a, (p - 1) / 2, p) != 1 { return 0 }
if p % 4 == 3 {
return modpow(a, (p + 1) / 4, p)
}
let mut q: i64 = p - 1
let mut s: i32 = 0
while q % 2 == 0 {
q = q / 2
s = s + 1
}
let mut z: i64 = 2
while modpow(z, (p - 1) / 2, p) != p - 1 {
z = z + 1
}
let mut m: i32 = s
let mut c: i64 = modpow(z, q, p)
let mut t: i64 = modpow(a, q, p)
let mut r: i64 = modpow(a, (q + 1) / 2, p)
while t != 1 {
let mut i: i32 = 1
let mut t2: i64 = (t * t) % p
while t2 != 1 {
t2 = (t2 * t2) % p
i = i + 1
}
let mut e: i32 = m - i - 1
let mut b: i64 = c
let mut j: i32 = 0
while j < e {
b = (b * b) % p
j = j + 1
}
r = (r * b) % p
t = (((t * b) % p) * b) % p
c = (b * b) % p
m = i
}
return r
}
function main() -> i32 {
let LIMIT: i64 = 50000000
let max_value: i64 = 2 * LIMIT * LIMIT - 1
let plim: i64 = isqrt(max_value) + 1
let sieve: ptr<i8> = calloc(plim + 1, 1)
if sieve == null { return 1 }
let mut i: i64 = 0
while i <= plim {
sieve[i] = 1
i = i + 1
}
sieve[0] = 0
sieve[1] = 0
i = 2
while i * i <= plim {
if sieve[i] == 1 {
let mut j: i64 = i * i
while j <= plim {
sieve[j] = 0
j = j + i
}
}
i = i + 1
}
let composite: ptr<i8> = calloc(LIMIT + 1, 1)
if composite == null { return 1 }
i = 3
while i <= plim {
if sieve[i] == 1 {
let rem: i64 = i & 7
if rem == 1 || rem == 7 {
let inv2: i64 = (i + 1) / 2
let mut root: i64 = 0
if rem == 7 {
root = (modpow(2, (i + 1) / 4, i) * inv2) % i
} else {
root = tonelli(inv2, i)
}
if root != 0 || rem == 7 {
# mark root, -root progressions
let mut start: i64 = root
if start < 2 {
start = start + ((2 - start + i - 1) / i) * i
}
let mut j: i64 = start
while j <= LIMIT {
composite[j] = 1
j = j + i
}
let other: i64 = (i - root) % i
if other != root {
start = other
if start < 2 {
start = start + ((2 - start + i - 1) / i) * i
}
j = start
while j <= LIMIT {
composite[j] = 1
j = j + i
}
}
let maybe: i64 = isqrt(inv2)
if maybe * maybe == inv2 && maybe <= LIMIT {
composite[maybe] = 0
}
}
}
}
i = i + 1
}
let mut count: i64 = 0
i = 2
while i <= LIMIT {
if composite[i] == 0 { count = count + 1 }
i = i + 1
}
printf("%lld\n", count)
free(sieve); free(composite)
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);
int64_t modpow_i64_i64_i64(int64_t base, int64_t exp, int64_t mod);
int64_t tonelli_i64_i64(int64_t a0, int64_t p);
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;
}
int64_t modpow_i64_i64_i64(int64_t base, int64_t exp, int64_t mod) {
int64_t r = 1;
int64_t b = FLOW_CHECKED_MOD((base), (mod));
int64_t e = exp;
while (e > 0) {
if (FLOW_CHECKED_MOD((e), (2)) == 1) {
r = FLOW_CHECKED_MOD(((r * b)), (mod));
}
b = FLOW_CHECKED_MOD(((b * b)), (mod));
e = FLOW_CHECKED_DIV((e), (2));
}
return r;
}
int64_t tonelli_i64_i64(int64_t a0, int64_t p) {
int64_t a = FLOW_CHECKED_MOD((a0), (p));
if (a == 0) {
return 0;
}
if (modpow_i64_i64_i64(a, FLOW_CHECKED_DIV(((p - 1)), (2)), p) != 1) {
return 0;
}
if (FLOW_CHECKED_MOD((p), (4)) == 3) {
return modpow_i64_i64_i64(a, FLOW_CHECKED_DIV(((p + 1)), (4)), p);
}
int64_t q = (p - 1);
int32_t s = 0;
while (FLOW_CHECKED_MOD((q), (2)) == 0) {
q = FLOW_CHECKED_DIV((q), (2));
s = (s + 1);
}
int64_t z = 2;
while (modpow_i64_i64_i64(z, FLOW_CHECKED_DIV(((p - 1)), (2)), p) != (p - 1)) {
z = (z + 1);
}
int32_t m = s;
int64_t c = modpow_i64_i64_i64(z, q, p);
int64_t t = modpow_i64_i64_i64(a, q, p);
int64_t r = modpow_i64_i64_i64(a, FLOW_CHECKED_DIV(((q + 1)), (2)), p);
while (t != 1) {
int32_t i = 1;
int64_t t2 = FLOW_CHECKED_MOD(((t * t)), (p));
while (t2 != 1) {
t2 = FLOW_CHECKED_MOD(((t2 * t2)), (p));
i = (i + 1);
}
int32_t e = ((m - i) - 1);
int64_t b = c;
int32_t j = 0;
while (j < e) {
b = FLOW_CHECKED_MOD(((b * b)), (p));
j = (j + 1);
}
r = FLOW_CHECKED_MOD(((r * b)), (p));
t = FLOW_CHECKED_MOD(((FLOW_CHECKED_MOD(((t * b)), (p)) * b)), (p));
c = FLOW_CHECKED_MOD(((b * b)), (p));
m = i;
}
return r;
}
int32_t main(void) {
int64_t LIMIT = 50000000;
int64_t max_value = (((2 * LIMIT) * LIMIT) - 1);
int64_t plim = (isqrt_i64(max_value) + 1);
int8_t* sieve = (int8_t*)(calloc((plim + 1), 1));
if (sieve == NULL) {
return 1;
}
int64_t i = 0;
while (i <= plim) {
sieve[i] = 1;
i = (i + 1);
}
sieve[0] = 0;
sieve[1] = 0;
i = 2;
while ((i * i) <= plim) {
if (sieve[i] == 1) {
int64_t j = (i * i);
while (j <= plim) {
sieve[j] = 0;
j = (j + i);
}
}
i = (i + 1);
}
int8_t* composite = (int8_t*)(calloc((LIMIT + 1), 1));
if (composite == NULL) {
return 1;
}
i = 3;
while (i <= plim) {
if (sieve[i] == 1) {
int64_t rem = (i & 7);
if ((rem == 1 || rem == 7)) {
int64_t inv2 = FLOW_CHECKED_DIV(((i + 1)), (2));
int64_t root = 0;
if (rem == 7) {
root = FLOW_CHECKED_MOD(((modpow_i64_i64_i64(2, FLOW_CHECKED_DIV(((i + 1)), (4)), i) * inv2)), (i));
} else {
root = tonelli_i64_i64(inv2, i);
}
if ((root != 0 || rem == 7)) {
int64_t start = root;
if (start < 2) {
start = (start + (FLOW_CHECKED_DIV(((((2 - start) + i) - 1)), (i)) * i));
}
int64_t j = start;
while (j <= LIMIT) {
composite[j] = 1;
j = (j + i);
}
int64_t other = FLOW_CHECKED_MOD(((i - root)), (i));
if (other != root) {
start = other;
if (start < 2) {
start = (start + (FLOW_CHECKED_DIV(((((2 - start) + i) - 1)), (i)) * i));
}
j = start;
while (j <= LIMIT) {
composite[j] = 1;
j = (j + i);
}
}
int64_t maybe = isqrt_i64(inv2);
if (((maybe * maybe) == inv2 && maybe <= LIMIT)) {
composite[maybe] = 0;
}
}
}
}
i = (i + 1);
}
int64_t count = 0;
i = 2;
while (i <= LIMIT) {
if (composite[i] == 0) {
count = (count + 1);
}
i = (i + 1);
}
printf("%lld\n", count);
free(sieve);
free(composite);
return 0;
}