# Project Euler 952
# Order Modulo Factorial. Multiplicative order of prime p modulo n!.
# Compute R(10^9+7, 10^7) mod 10^9+7.
extern {
function calloc(n: i64, size: i64) -> ptr<void>
function free(p: ptr<void>) -> void
}
const P: i64 = 1000000007
const N: i64 = 10000000
function mod_pow_u64(a0: i64, e0: i64, mod: i64) -> i64 {
let mut r: i64 = 1 % mod
let mut a: i64 = a0 % mod
let mut e: i64 = e0
while e > 0 {
if (e & 1) == 1 {
r = ((r as i128 * (a as i128)) % (mod as i128)) as i64
}
a = ((a as i128 * (a as i128)) % (mod as i128)) as i64
e = e >> 1
}
return r
}
function v_p_factorial(n0: i64, p: i64) -> i64 {
let mut s: i64 = 0
let mut n: i64 = n0
while n > 0 {
n = n / p
s = s + n
}
return s
}
function v2_val(x0: i64) -> i64 {
let mut c: i64 = 0
let mut x: i64 = x0
while (x & 1) == 0 {
x = x >> 1
c = c + 1
}
return c
}
function order_mod_2_power_exponent(a: i64, k: i64) -> i64 {
if k <= 1 { return 0 }
if k == 2 {
if (a & 3) == 1 { return 0 }
return 1
}
if (a & 3) == 1 {
let t: i64 = k - v2_val(a - 1)
if t > 0 { return t }
return 0
} else {
let t: i64 = k - v2_val(a + 1)
if t > 1 { return t }
return 1
}
}
# Global arrays
let mut spf: ptr<i32> = null
let mut primes: ptr<i32> = null
let mut num_primes: i64 = 0
let mut max_exp: ptr<i64> = null
function linear_sieve_spf(limit: i64) -> void {
spf = calloc(limit + 1, 4)
primes = calloc(700000, 4)
num_primes = 0
let mut i: i64 = 2
while i <= limit {
if spf[i] == 0 {
spf[i] = i as i32
primes[num_primes] = i as i32
num_primes = num_primes + 1
}
let mut j: i64 = 0
while j < num_primes {
let p: i64 = primes[j] as i64
let ip: i64 = i * p
if ip > limit { break }
spf[ip] = p as i32
if p == (spf[i] as i64) { break }
j = j + 1
}
i = i + 1
}
}
# Factorize x using SPF. Returns number of (prime, exponent) pairs.
function factorize(x0: i64, fac_p: ptr<i64>, fac_e: ptr<i64>) -> i64 {
let mut cnt: i64 = 0
let mut x: i64 = x0
while x > 1 {
let p: i64 = spf[x] as i64
let mut e: i64 = 0
while x % p == 0 {
x = x / p
e = e + 1
}
fac_p[cnt] = p
fac_e[cnt] = e
cnt = cnt + 1
}
return cnt
}
function multiplicative_order_mod_prime(p: i64, q: i64) -> i64 {
let m: i64 = q - 1
let fac_p: ptr<i64> = calloc(32, 8)
let fac_e: ptr<i64> = calloc(32, 8)
let nfac: i64 = factorize(m, fac_p, fac_e)
let mut r: i64 = m
let base: i64 = p % q
let mut i: i64 = 0
while i < nfac {
let f: i64 = fac_p[i]
let mut e_rem: i64 = fac_e[i]
while e_rem > 0 {
let cand: i64 = r / f
if mod_pow_u64(base, cand, q) == 1 {
r = cand
e_rem = e_rem - 1
} else {
break
}
}
if e_rem > max_exp[f] {
max_exp[f] = e_rem
}
i = i + 1
}
free(fac_p)
free(fac_e)
return r
}
function q_adic_valuation(p: i64, r: i64, q: i64, limit: i64) -> i64 {
if limit <= 1 { return 1 }
let mut s: i64 = 1
let mut q_pow: i64 = q
while s < limit {
let q_pow_next: i64 = q_pow * q
if mod_pow_u64(p, r, q_pow_next) != 1 { break }
q_pow = q_pow_next
s = s + 1
}
return s
}
function main() -> i32 {
linear_sieve_spf(N)
max_exp = calloc(N + 1, 8)
# Handle q=2 separately
let a2: i64 = v_p_factorial(N, 2)
let t2: i64 = order_mod_2_power_exponent(P, a2)
if t2 > max_exp[2] { max_exp[2] = t2 }
# Odd primes
let mut qi: i64 = 0
while qi < num_primes {
let q: i64 = primes[qi] as i64
if q == 2 {
qi = qi + 1
continue
}
let a: i64 = v_p_factorial(N, q)
let r0: i64 = multiplicative_order_mod_prime(P, q)
let s: i64 = q_adic_valuation(P, r0, q, a)
let extra: i64 = a - s
if extra > 0 && extra > max_exp[q] {
max_exp[q] = extra
}
qi = qi + 1
}
# Reconstruct order modulo P
let mut res: i64 = 1 % P
let mut qi2: i64 = 0
while qi2 < num_primes {
let q: i64 = primes[qi2] as i64
let e: i64 = max_exp[q]
if e != 0 {
res = ((res as i128 * (mod_pow_u64(q, e, P) as i128)) % (P as i128)) as i64
}
qi2 = qi2 + 1
}
printf("%lld\n", res)
free(spf)
free(primes)
free(max_exp)
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 mod_pow_u64_i64_i64_i64(int64_t a0, int64_t e0, int64_t mod);
int64_t v_p_factorial_i64_i64(int64_t n0, int64_t p);
int64_t v2_val_i64(int64_t x0);
int64_t order_mod_2_power_exponent_i64_i64(int64_t a, int64_t k);
void linear_sieve_spf_i64(int64_t limit);
int64_t factorize_i64_ptr_i64_ptr_i64(int64_t x0, int64_t* fac_p, int64_t* fac_e);
int64_t multiplicative_order_mod_prime_i64_i64(int64_t p, int64_t q);
int64_t q_adic_valuation_i64_i64_i64_i64(int64_t p, int64_t r, int64_t q, int64_t limit);
int32_t main(void);
static const int64_t P = 1000000007;
static const int64_t N = 10000000;
/* Module statics */
static int32_t* spf = NULL;
static int32_t* primes = NULL;
static int64_t num_primes = 0;
static int64_t* max_exp = NULL;
int64_t mod_pow_u64_i64_i64_i64(int64_t a0, int64_t e0, int64_t mod) {
int64_t r = FLOW_CHECKED_MOD((1), (mod));
int64_t a = FLOW_CHECKED_MOD((a0), (mod));
int64_t e = e0;
while (e > 0) {
if ((e & 1) == 1) {
r = ((int64_t)(FLOW_CHECKED_MOD(((((__int128)(r)) * ((__int128)(a)))), (((__int128)(mod))))));
}
a = ((int64_t)(FLOW_CHECKED_MOD(((((__int128)(a)) * ((__int128)(a)))), (((__int128)(mod))))));
e = FLOW_CHECKED_SHR((e), (1));
}
return r;
}
int64_t v_p_factorial_i64_i64(int64_t n0, int64_t p) {
int64_t s = 0;
int64_t n = n0;
while (n > 0) {
n = FLOW_CHECKED_DIV((n), (p));
s = (s + n);
}
return s;
}
int64_t v2_val_i64(int64_t x0) {
int64_t c = 0;
int64_t x = x0;
while ((x & 1) == 0) {
x = FLOW_CHECKED_SHR((x), (1));
c = (c + 1);
}
return c;
}
int64_t order_mod_2_power_exponent_i64_i64(int64_t a, int64_t k) {
if (k <= 1) {
return 0;
}
if (k == 2) {
if ((a & 3) == 1) {
return 0;
}
return 1;
}
if ((a & 3) == 1) {
int64_t t = (k - v2_val_i64((a - 1)));
if (t > 0) {
return t;
}
return 0;
} else {
int64_t t = (k - v2_val_i64((a + 1)));
if (t > 1) {
return t;
}
return 1;
}
}
void linear_sieve_spf_i64(int64_t limit) {
spf = calloc((limit + 1), 4);
primes = calloc(700000, 4);
num_primes = 0;
int64_t i = 2;
while (i <= limit) {
if (spf[i] == 0) {
spf[i] = ((int32_t)(i));
primes[num_primes] = ((int32_t)(i));
num_primes = (num_primes + 1);
}
int64_t j = 0;
while (j < num_primes) {
int64_t p = ((int64_t)(primes[j]));
int64_t ip = (i * p);
if (ip > limit) {
break;
}
spf[ip] = ((int32_t)(p));
if (p == ((int64_t)(spf[i]))) {
break;
}
j = (j + 1);
}
i = (i + 1);
}
}
int64_t factorize_i64_ptr_i64_ptr_i64(int64_t x0, int64_t* fac_p, int64_t* fac_e) {
int64_t cnt = 0;
int64_t x = x0;
while (x > 1) {
int64_t p = ((int64_t)(spf[x]));
int64_t e = 0;
while (FLOW_CHECKED_MOD((x), (p)) == 0) {
x = FLOW_CHECKED_DIV((x), (p));
e = (e + 1);
}
fac_p[cnt] = p;
fac_e[cnt] = e;
cnt = (cnt + 1);
}
return cnt;
}
int64_t multiplicative_order_mod_prime_i64_i64(int64_t p, int64_t q) {
int64_t m = (q - 1);
int64_t* fac_p = (int64_t*)(calloc(32, 8));
int64_t* fac_e = (int64_t*)(calloc(32, 8));
int64_t nfac = factorize_i64_ptr_i64_ptr_i64(m, fac_p, fac_e);
int64_t r = m;
int64_t base = FLOW_CHECKED_MOD((p), (q));
int64_t i = 0;
while (i < nfac) {
int64_t f = fac_p[i];
int64_t e_rem = fac_e[i];
while (e_rem > 0) {
int64_t cand = FLOW_CHECKED_DIV((r), (f));
if (mod_pow_u64_i64_i64_i64(base, cand, q) == 1) {
r = cand;
e_rem = (e_rem - 1);
} else {
break;
}
}
if (e_rem > max_exp[f]) {
max_exp[f] = e_rem;
}
i = (i + 1);
}
free(fac_p);
free(fac_e);
return r;
}
int64_t q_adic_valuation_i64_i64_i64_i64(int64_t p, int64_t r, int64_t q, int64_t limit) {
if (limit <= 1) {
return 1;
}
int64_t s = 1;
int64_t q_pow = q;
while (s < limit) {
int64_t q_pow_next = (q_pow * q);
if (mod_pow_u64_i64_i64_i64(p, r, q_pow_next) != 1) {
break;
}
q_pow = q_pow_next;
s = (s + 1);
}
return s;
}
int32_t main(void) {
linear_sieve_spf_i64(N);
max_exp = calloc((N + 1), 8);
int64_t a2 = v_p_factorial_i64_i64(N, 2);
int64_t t2 = order_mod_2_power_exponent_i64_i64(P, a2);
if (t2 > max_exp[2]) {
max_exp[2] = t2;
}
int64_t qi = 0;
while (qi < num_primes) {
int64_t q = ((int64_t)(primes[qi]));
if (q == 2) {
qi = (qi + 1);
continue;
}
int64_t a = v_p_factorial_i64_i64(N, q);
int64_t r0 = multiplicative_order_mod_prime_i64_i64(P, q);
int64_t s = q_adic_valuation_i64_i64_i64_i64(P, r0, q, a);
int64_t extra = (a - s);
if ((extra > 0 && extra > max_exp[q])) {
max_exp[q] = extra;
}
qi = (qi + 1);
}
int64_t res = FLOW_CHECKED_MOD((1), (P));
int64_t qi2 = 0;
while (qi2 < num_primes) {
int64_t q = ((int64_t)(primes[qi2]));
int64_t e = max_exp[q];
if (e != 0) {
res = ((int64_t)(FLOW_CHECKED_MOD(((((__int128)(res)) * ((__int128)(mod_pow_u64_i64_i64_i64(q, e, P))))), (((__int128)(P))))));
}
qi2 = (qi2 + 1);
}
printf("%lld\n", res);
free(spf);
free(primes);
free(max_exp);
return 0;
}