Convex Path in Square: F(10^18). Maximize lattice points on a strictly convex increasing curve in N×N square. Use primitive step vectors (a,b) with gcd(a,b)=1, grouped by s=a+b. Count = φ(s) per group, width/height = s*φ(s)/2. Greedy + remainder.
# Project Euler 604
# Convex Path in Square: F(10^18).
# Maximize lattice points on a strictly convex increasing curve in N×N square.
# Use primitive step vectors (a,b) with gcd(a,b)=1, grouped by s=a+b.
# Count = φ(s) per group, width/height = s*φ(s)/2. Greedy + remainder.
import euler.nt { gcd, isqrt }
extern {
function calloc(n: i64, size: i64) -> ptr<void>
function free(p: ptr<void>) -> void
}
function iroot3(n: i64) -> i64 {
let mut lo: i64 = 0
let mut hi: i64 = 2
while hi * hi * hi <= n { hi = hi * 2 }
while lo + 1 < hi {
let mid: i64 = (lo + hi) / 2
if mid * mid * mid <= n { lo = mid }
elif mid * mid * mid > n { hi = mid }
}
return lo
}
function main() -> i32 {
let N: i64 = 1000000000000000000
# M ≈ (10N)^(1/3) + margin, but 10*N overflows i64 so use iroot3(N)*3
let mut M: i64 = iroot3(N) * 3 + 1000
if M < 10 { M = 10 }
# Linear sieve for totients up to M+1
let phi: ptr<i64> = calloc(M + 2, 8)
if phi == null { return 1 }
let is_comp: ptr<i8> = calloc(M + 2, 1)
if is_comp == null { return 1 }
let primes: ptr<i32> = calloc(M / 3 + 10, 4)
if primes == null { return 1 }
phi[1] = 1
let mut pc: i64 = 0
for i in 2..(M + 2) {
if is_comp[i] == 0 {
primes[pc] = i as i32
pc = pc + 1
phi[i] = i - 1
}
for j in 0..pc {
let p: i64 = primes[j] as i64
let ip: i64 = i * p
if ip > M + 1 { break }
is_comp[ip] = 1
if i % p == 0 { phi[ip] = phi[i] * p; break }
elif i % p != 0 { phi[ip] = phi[i] * (p - 1) }
}
}
# Find max t with sum_{s=2..t} s*phi(s) <= 2*N
let target: i64 = 2 * N
let mut sum_sphi: i64 = 0
let mut t: i64 = 1
let mut base_phi: i64 = 0
let mut base_sphi: i64 = 0
for i in 2..(M + 1) {
let sp: i64 = i * phi[i]
if sum_sphi + sp > target { break }
sum_sphi = sum_sphi + sp
t = i
base_phi = base_phi + phi[i]
base_sphi = sum_sphi
}
let R: i64 = N - base_sphi / 2
let s0: i64 = t + 1
let p_count: i64 = R / s0
let mut add: i64 = 2 * p_count
let r: i64 = R - p_count * s0
# Try to add one more single vector from layer s0
if r > 0 {
for a in 1..s0 {
if gcd(a, s0) == 1 && a <= r && (s0 - a) <= r {
add = add + 1
break
}
}
}
printf("%lld\n", base_phi + add + 1)
free(primes)
free(is_comp)
free(phi)
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 iroot3_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;
}
int64_t iroot3_i64(int64_t n) {
int64_t lo = 0;
int64_t hi = 2;
while (((hi * hi) * hi) <= n) {
hi = (hi * 2);
}
while ((lo + 1) < hi) {
int64_t mid = FLOW_CHECKED_DIV(((lo + hi)), (2));
if (((mid * mid) * mid) <= n) {
lo = mid;
} else if (((mid * mid) * mid) > n) {
hi = mid;
}
}
return lo;
}
int32_t main(void) {
int64_t N = 1000000000000000000;
int64_t M = ((iroot3_i64(N) * 3) + 1000);
if (M < 10) {
M = 10;
}
int64_t* phi = (int64_t*)(calloc((M + 2), 8));
if (phi == NULL) {
return 1;
}
int8_t* is_comp = (int8_t*)(calloc((M + 2), 1));
if (is_comp == NULL) {
return 1;
}
int32_t* primes = (int32_t*)(calloc((FLOW_CHECKED_DIV((M), (3)) + 10), 4));
if (primes == NULL) {
return 1;
}
phi[1] = 1;
int64_t pc = 0;
int32_t __flow_step_1 = 1;
for (int32_t i = 2; (2 <= (M + 2)) ? i < (M + 2) : i > (M + 2); i += (2 <= (M + 2)) ? 1 : -1) {
if (is_comp[i] == 0) {
primes[pc] = ((int32_t)(i));
pc = (pc + 1);
phi[i] = (i - 1);
}
int32_t __flow_step_2 = 1;
for (int32_t j = 0; (0 <= pc) ? j < pc : j > pc; j += (0 <= pc) ? 1 : -1) {
int64_t p = ((int64_t)(primes[j]));
int64_t ip = (i * p);
if (ip > (M + 1)) {
break;
}
is_comp[ip] = 1;
if (FLOW_CHECKED_MOD((i), (p)) == 0) {
phi[ip] = (phi[i] * p);
break;
} else if (FLOW_CHECKED_MOD((i), (p)) != 0) {
phi[ip] = (phi[i] * (p - 1));
}
}
}
int64_t target = (2 * N);
int64_t sum_sphi = 0;
int64_t t = 1;
int64_t base_phi = 0;
int64_t base_sphi = 0;
int32_t __flow_step_3 = 1;
for (int32_t i = 2; (2 <= (M + 1)) ? i < (M + 1) : i > (M + 1); i += (2 <= (M + 1)) ? 1 : -1) {
int64_t sp = (i * phi[i]);
if ((sum_sphi + sp) > target) {
break;
}
sum_sphi = (sum_sphi + sp);
t = i;
base_phi = (base_phi + phi[i]);
base_sphi = sum_sphi;
}
int64_t R = (N - FLOW_CHECKED_DIV((base_sphi), (2)));
int64_t s0 = (t + 1);
int64_t p_count = FLOW_CHECKED_DIV((R), (s0));
int64_t add = (2 * p_count);
int64_t r = (R - (p_count * s0));
if (r > 0) {
int32_t __flow_step_4 = 1;
for (int32_t a = 1; (1 <= s0) ? a < s0 : a > s0; a += (1 <= s0) ? 1 : -1) {
if (((gcd_i64_i64(a, s0) == 1 && a <= r) && (s0 - a) <= r)) {
add = (add + 1);
break;
}
}
}
printf("%lld\n", ((base_phi + add) + 1));
free(primes);
free(is_comp);
free(phi);
return 0;
}