# Project Euler 188
# Last 8 digits of 1777↑↑1855.
import euler.nt { gcd }
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 carmichael(n: i64) -> i64 {
# λ(n) for n = 2^a * odd
let mut m: i64 = n
let mut lam: i64 = 1
if m % 2 == 0 {
let mut a: i64 = 0
while m % 2 == 0 {
m = m / 2
a = a + 1
}
if a == 1 { lam = 1 }
elif a == 2 { lam = 2 }
else {
let mut t: i64 = 1
for i in 0..(a - 2) {
t = t * 2
}
lam = t
}
}
let mut p: i64 = 3
while p * p <= m {
if m % p == 0 {
let mut pk: i64 = 1
while m % p == 0 {
m = m / p
pk = pk * p
}
let phi_pk: i64 = pk - pk / p
# lcm(lam, phi_pk)
lam = lam / gcd(lam, phi_pk) * phi_pk
}
p = p + 2
}
if m > 1 {
let phi_m: i64 = m - 1
lam = lam / gcd(lam, phi_m) * phi_m
}
return lam
}
function tower(a: i64, h: i64, mod: i64) -> i64 {
if mod == 1 { return 0 }
if h == 1 { return a % mod }
if h == 2 { return modpow(a, a, mod) }
let lam: i64 = carmichael(mod)
let e: i64 = tower(a, h - 1, lam)
return modpow(a, e, mod)
}
function main() -> i32 {
# Totient chain for 10^8 is short; height 1855 collapses quickly.
# Cap height: once mod's carmichael chain depth < h, same result.
let mut h: i64 = 1855
if h > 40 { h = 40 } # safe: λ-chain of 10^8 << 40
printf("%lld\n", tower(1777, h, 100000000))
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 carmichael_i64(int64_t n);
int64_t tower_i64_i64_i64(int64_t a, int64_t h, int64_t mod);
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 carmichael_i64(int64_t n) {
int64_t m = n;
int64_t lam = 1;
if (FLOW_CHECKED_MOD((m), (2)) == 0) {
int64_t a = 0;
while (FLOW_CHECKED_MOD((m), (2)) == 0) {
m = FLOW_CHECKED_DIV((m), (2));
a = (a + 1);
}
if (a == 1) {
lam = 1;
} else if (a == 2) {
lam = 2;
} else {
int64_t t = 1;
int32_t __flow_step_1 = 1;
for (int32_t i = 0; (0 <= (a - 2)) ? i < (a - 2) : i > (a - 2); i += (0 <= (a - 2)) ? 1 : -1) {
t = (t * 2);
}
lam = t;
}
}
int64_t p = 3;
while ((p * p) <= m) {
if (FLOW_CHECKED_MOD((m), (p)) == 0) {
int64_t pk = 1;
while (FLOW_CHECKED_MOD((m), (p)) == 0) {
m = FLOW_CHECKED_DIV((m), (p));
pk = (pk * p);
}
int64_t phi_pk = (pk - FLOW_CHECKED_DIV((pk), (p)));
lam = (FLOW_CHECKED_DIV((lam), (gcd_i64_i64(lam, phi_pk))) * phi_pk);
}
p = (p + 2);
}
if (m > 1) {
int64_t phi_m = (m - 1);
lam = (FLOW_CHECKED_DIV((lam), (gcd_i64_i64(lam, phi_m))) * phi_m);
}
return lam;
}
int64_t tower_i64_i64_i64(int64_t a, int64_t h, int64_t mod) {
if (mod == 1) {
return 0;
}
if (h == 1) {
return FLOW_CHECKED_MOD((a), (mod));
}
if (h == 2) {
return modpow_i64_i64_i64(a, a, mod);
}
int64_t lam = carmichael_i64(mod);
int64_t e = tower_i64_i64_i64(a, (h - 1), lam);
return modpow_i64_i64_i64(a, e, mod);
}
int32_t main(void) {
int64_t h = 1855;
if (h > 40) {
h = 40;
}
printf("%lld\n", tower_i64_i64_i64(1777, h, 100000000));
return 0;
}