Bug in range sieve when left boundry is 0

nartherion · Volodymyr Sachko · commit 20f26762d01c · 2025-06-15T14:21:05.000+02:00
In case when `L = 0` and `R &gt; 0`, algorithm will say that 0 and 1 are primes.

Index initialization improvement
diff --git a/src/algebra/sieve-of-eratosthenes.md b/src/algebra/sieve-of-eratosthenes.md
@@ -207,7 +207,7 @@ To solve such a problem, we can use the idea of the Segmented sieve.
 We pre-generate all prime numbers up to $\sqrt R$, and use those primes to mark all composite numbers in the segment $[L, R]$.
 
 ```cpp
-vector<char> segmentedSieve(long long L, long long R) {
+vector<char> segmented_sieve(long long L, long long R) {
     // generate all primes up to sqrt(R)
     long long lim = sqrt(R);
     vector<char> mark(lim + 1, false);
@@ -222,10 +222,12 @@ vector<char> segmentedSieve(long long L, long long R) {
 
     vector<char> isPrime(R - L + 1, true);
     for (long long i : primes)
-        for (long long j = max(i * i, (L + i - 1) / i * i); j <= R; j += i)
+        for (long long j = max(i, (L + i - 1) / i) * i; j <= R; j += i)
             isPrime[j - L] = false;
-    if (L == 1)
-        isPrime[0] = false;
+    if (L == 0)
+        isPrime[L] = false;
+    if (L <= 1)
+        isPrime[min(1 - L, R - L)] = false;
     return isPrime;
 }
 ```
@@ -234,14 +236,16 @@ Time complexity of this approach is $O((R - L + 1) \log \log (R) + \sqrt R \log
 It's also possible that we don't pre-generate all prime numbers:
 
 ```cpp
-vector<char> segmentedSieveNoPreGen(long long L, long long R) {
+vector<char> segmented_sieve_no_pre_gen(long long L, long long R) {
     vector<char> isPrime(R - L + 1, true);
     long long lim = sqrt(R);
     for (long long i = 2; i <= lim; ++i)
-        for (long long j = max(i * i, (L + i - 1) / i * i); j <= R; j += i)
+        for (long long j = max(i, (L + i - 1) / i) * i; j <= R; j += i)
             isPrime[j - L] = false;
-    if (L == 1)
-        isPrime[0] = false;
+    if (L == 0)
+        isPrime[L] = false;
+    if (L <= 1)
+        isPrime[min(1 - L, R - L)] = false;
     return isPrime;
 }
 ```
@@ -258,6 +262,7 @@ However, this algorithm also has its own weaknesses.
 
 * [Leetcode - Four Divisors](https://leetcode.com/problems/four-divisors/)
 * [Leetcode - Count Primes](https://leetcode.com/problems/count-primes/)
+* [Leetcode - Closest Prime Numbers in Range](https://leetcode.com/problems/closest-prime-numbers-in-range/)
 * [SPOJ - Printing Some Primes](http://www.spoj.com/problems/TDPRIMES/)
 * [SPOJ - A Conjecture of Paul Erdos](http://www.spoj.com/problems/HS08PAUL/)
 * [SPOJ - Primal Fear](http://www.spoj.com/problems/VECTAR8/)
