# Project Euler 656
# Palindromic Sequences: sum of H_100(sqrt(beta)) for non-square beta in 2..1000.
# Uses continued fraction periods and palindromic prefix length formula.
import euler.nt { isqrt }
# Compute the continued fraction period of sqrt(n) for non-square n.
# Returns period length in *plen, period coefficients in *period.
function cf_period(n: i64, period: ptr<i64>) -> i64 {
let a0: i64 = isqrt(n)
let mut m: i64 = 0
let mut d: i64 = 1
let mut a: i64 = a0
let mut len: i64 = 0
while true {
m = d * a - m
d = (n - m * m) / d
a = (a0 + m) / d
period[len] = a
len = len + 1
if a == 2 * a0 { break }
}
return len
}
# Compute H_g(sqrt(beta)) mod 10^15 using the continued fraction period.
# Palindromic prefix lengths: n = q_{k-2} + t * q_{k-1} for odd k, 1 <= t <= a_k.
function H_mod(period: ptr<i64>, plen: i64, g: i64, mod: i64) -> i64 {
let mut q_m1: i64 = 0
let mut q0: i64 = 1
let mut k: i64 = 1
let mut i: i64 = 0
let mut count: i64 = 0
let mut s: i64 = 0
while count < g {
let a: i64 = period[i]
i = i + 1
if i == plen { i = 0 }
if k % 2 == 1 {
let remaining: i64 = g - count
let tmax: i64 = a
if tmax > remaining { tmax = remaining }
# sum_{t=1..tmax} (q_m1 + t*q0) = tmax*q_m1 + q0*tmax*(tmax+1)/2
let t1: i128 = ((tmax % mod) as i128) * ((q_m1 % mod) as i128) % (mod as i128)
s = (s + (t1 as i64)) % mod
let tri_val: i64 = (tmax * (tmax + 1) / 2) % mod
let t2: i128 = ((q0 % mod) as i128) * (tri_val as i128) % (mod as i128)
s = (s + (t2 as i64)) % mod
count = count + tmax
}
let q1: i64 = (((a as i128) * (q0 as i128) + (q_m1 as i128)) % (mod as i128)) as i64
q_m1 = q0
q0 = q1
k = k + 1
}
return s % mod
}
function main() -> i32 {
let MOD: i64 = 1000000000000000
let LIMIT_BETA: i64 = 1000
let G: i64 = 100
let period: ptr<i64> = calloc(256, 8)
if period == null { return 1 }
let mut total: i64 = 0
for beta in 2..(LIMIT_BETA + 1) {
let r: i64 = isqrt(beta)
if r * r == beta { continue }
let plen: i64 = cf_period(beta, period)
let h: i64 = H_mod(period, plen, G, MOD)
total = (total + h) % MOD
if total < 0 { total = total + MOD }
}
printf("%015lld\n", total)
free(period)
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 cf_period_i64_ptr_i64(int64_t n, int64_t* period);
int64_t H_mod_ptr_i64_i64_i64_i64(int64_t* period, int64_t plen, int64_t g, 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 cf_period_i64_ptr_i64(int64_t n, int64_t* period) {
int64_t a0 = isqrt_i64(n);
int64_t m = 0;
int64_t d = 1;
int64_t a = a0;
int64_t len = 0;
while (1) {
m = ((d * a) - m);
d = FLOW_CHECKED_DIV(((n - (m * m))), (d));
a = FLOW_CHECKED_DIV(((a0 + m)), (d));
period[len] = a;
len = (len + 1);
if (a == (2 * a0)) {
break;
}
}
return len;
}
int64_t H_mod_ptr_i64_i64_i64_i64(int64_t* period, int64_t plen, int64_t g, int64_t mod) {
int64_t q_m1 = 0;
int64_t q0 = 1;
int64_t k = 1;
int64_t i = 0;
int64_t count = 0;
int64_t s = 0;
while (count < g) {
int64_t a = period[i];
i = (i + 1);
if (i == plen) {
i = 0;
}
if (FLOW_CHECKED_MOD((k), (2)) == 1) {
int64_t remaining = (g - count);
int64_t tmax = a;
if (tmax > remaining) {
tmax = remaining;
}
__int128 t1 = FLOW_CHECKED_MOD(((((__int128)(FLOW_CHECKED_MOD((tmax), (mod)))) * ((__int128)(FLOW_CHECKED_MOD((q_m1), (mod)))))), (((__int128)(mod))));
s = FLOW_CHECKED_MOD(((s + ((int64_t)(t1)))), (mod));
int64_t tri_val = FLOW_CHECKED_MOD((FLOW_CHECKED_DIV(((tmax * (tmax + 1))), (2))), (mod));
__int128 t2 = FLOW_CHECKED_MOD(((((__int128)(FLOW_CHECKED_MOD((q0), (mod)))) * ((__int128)(tri_val)))), (((__int128)(mod))));
s = FLOW_CHECKED_MOD(((s + ((int64_t)(t2)))), (mod));
count = (count + tmax);
}
int64_t q1 = ((int64_t)(FLOW_CHECKED_MOD((((((__int128)(a)) * ((__int128)(q0))) + ((__int128)(q_m1)))), (((__int128)(mod))))));
q_m1 = q0;
q0 = q1;
k = (k + 1);
}
return FLOW_CHECKED_MOD((s), (mod));
}
int32_t main(void) {
int64_t MOD = 1000000000000000;
int64_t LIMIT_BETA = 1000;
int64_t G = 100;
int64_t* period = (int64_t*)(calloc(256, 8));
if (period == NULL) {
return 1;
}
int64_t total = 0;
int32_t __flow_step_1 = 1;
for (int32_t beta = 2; (2 <= (LIMIT_BETA + 1)) ? beta < (LIMIT_BETA + 1) : beta > (LIMIT_BETA + 1); beta += (2 <= (LIMIT_BETA + 1)) ? 1 : -1) {
int64_t r = isqrt_i64(beta);
if ((r * r) == beta) {
continue;
}
int64_t plen = cf_period_i64_ptr_i64(beta, period);
int64_t h = H_mod_ptr_i64_i64_i64_i64(period, plen, G, MOD);
total = FLOW_CHECKED_MOD(((total + h)), (MOD));
if (total < 0) {
total = (total + MOD);
}
}
printf("%015lld\n", total);
free(period);
return 0;
}