Develop Reference

Algorithm to find prime numbers with odd digits in large range

Given a range of [1, 1000000000] we need to find prime numbers and all the digits of the prime number must be odd. (Example: 23 is not okay, 31 is okay)
If we go on by looping through each number and checking if it is prime etc, it is very slow. Is there a way to make this close to O(N) ?
I tried to eliminate as much as possible by looking at digits first. But after eliminating numbers with even digits the prime test is too slow.
for all numbers in [1, N]
check all digits, if any even digit, continue
check primality (this step is very slow)
And the primality test should not be very complex (probabilistic etc. is not possible, it must be possible to implement in a few minutes). The one I use is:
private static boolean isPrime(int n) {
boolean isPrime = true;
for (int divisor = 2; divisor <= n / 2; divisor++) {
if (n % divisor == 0) {
isPrime = false;
break;
}
}
return isPrime;
}
Maybe there is a trick to ensure a quick primality test but I couldn't find. Any suggestions? Thanks for reading.

You don't need to check all milliard numbers. Generate all numbers with only odd digits - there are at most 5^9~2 millions of them. Exclude those ending with 5 and not generate numbers divisible by 3 (in the moment of the last digit generation)
Then check these numbers for primality. Note that loop limit might be sqrt(n)
Ideone
class Ideone
{
static int oddcnt;
public static void checkprime(int x) {
for (int i=3; i <= Math.sqrt(x); i +=2)
if ((x % i) == 0)
return;
oddcnt++;
}
public static void genodd(int x, int curlen, int maxlen) {
x *= 10;
for (int i=1; i<10; i+=2) {
int nx = x + i;
checkprime(nx);
if (curlen < maxlen)
genodd(nx, curlen + 1, maxlen);
}
}
public static void main (String[] args) throws java.lang.Exception
{
genodd(0, 1, 8);
System.out.println(oddcnt);
}
}

The best way I can think of is to run a Prime Sieve of Eratosthenes to find all the primes in the range (0; sqrt(1000000000)) - which is around (0, 31622) - and time complexity O(n*log(log(n))) where n=31622. We will need those prime for a faster primality test.
Then, just loop through each number with odd digits - there are 5^10 = 9765625 ~ 10000000 such numbers. You saved 1000 times compared to iterating through all number in the original range.
The primality test using the primes we found in step 1 can be fast, as you only need to check with primes < sqrt(n), and you already have the primes. Even for the largest number in the range which is 999999999, the number of candidate primes is just 3432.
The following is a Java implementation
public class Execute {
private ArrayList<Long> primes = new ArrayList<>();
#org.junit.Test
public void findOddDecimalPrimes() {
primeSieve(32000);
System.out.println(primes.size());
for (int i = 0; i < 9765625; i++) {
String inBase5 = convertFromBaseToBase(i);
long evenDec = convertToOddDecimal(inBase5);
if (isPrime(evenDec)) {
System.out.println(evenDec);
}
}
}
private String convertFromBaseToBase(long i) {
return Long.toString(i, 5);
}
private long convertToOddDecimal(String str) {
StringBuilder s = new StringBuilder();
for (int i = 0; i < str.length(); i++) {
s.append(1 + 2 * Integer.parseInt("" + str.charAt(i)));
}
return Long.parseLong(s.toString());
}
private boolean isPrime(long n) {
for (int i = 0; i < primes.size(); i++) {
if (primes.get(i) * primes.get(i) > n) break;
long divisor = n / primes.get(i);
if (divisor * primes.get(i) == n) return false;
}
return true;
}
/**
* References: www.geeksforgeeks.org
*/
private void primeSieve(int n)
{
// Create a boolean array "prime[0..n]" and initialize
// all entries it as true. A value in prime[i] will
// finally be false if i is Not a prime, else true.
boolean prime[] = new boolean[n+1];
for(int i=0;i<n;i++)
prime[i] = true;
for(int p = 2; p*p <=n; p++)
{
// If prime[p] is not changed, then it is a prime
if(prime[p] == true)
{
// Update all multiples of p
for(int i = p*p; i <= n; i += p)
prime[i] = false;
}
}
for (int i = 2; i < prime.length; i++) {
if (prime[i]) this.primes.add(Long.valueOf(i));
}
}
}

If your numbers are in order, you can optimize your isPrime function.
Here is a js sample version.
var primes = [];
function checkDigits(n) {
while(n > 1) {
var d = n % 10;
if ( d % 2 == 0) { return false; }
n = parseInt(n/10,10);
}
return true;
}
function isPrime(n) {
for(var i = 1; i < primes.length; i++) {
if(n % primes[i] == 0) {
return false;
}
}
var lastPrime = primes.length > 2 ? primes[primes.length - 1] : 1;
var inc = 2;
for(var i = lastPrime + inc; i < Math.sqrt(n); i += inc) {
if(n % i == 0) {
return false;
}
}
primes.push(n);
return true;
}
for(var i = 1; i < 100; i++) {
if(checkDigits(i) && isPrime(i)) {
console.log(i);
}
}

This is an open question in math and computer science in general.
In a nutshell no, there is no know way to solve that problem in O(1) to get your loop running in O(N) over the whole range.
If you solve that, dont tell anyone, and go get rich by breaking most of the encryptions today that use large prime numbers.
What you could do though, is make the loop over the devisor a bit smaller by useing sqrt(n).
That will bring that inner loop down from O(N^2) to O(sqrt(N))
And the whole complexity from O(N^2) to O(N*sqrt(N))=O(N^(3/2))
Anotger optimization would be to check the odd digits first beforre doing the complex Prime calculation

Printing Prime Numbers to N value

This program is supposed to print all the prime numbers up to an int that you enter, for example:
Enter a Number:
20
2
3
5
7
11
13
17
19
I just cannot get my program to work, I really don't know what to do, so if someone could review it and try to fix it, that would be greatly appreciated, thanks.
import java.util.Scanner;
public class PrimeGenerator {
public static void main(String args[]) {
Scanner k = new Scanner(System.in);
System.out.println("Enter an integer");
int number = k.nextInt();
PrimeGenerator matt = new PrimeGenerator();
System.out.println(matt.nextPrime(number));
}
private int number;
public PrimeGenerator(int n) {
number = n;
}
public int nextPrime(int number) {
for (int i = 1; i <= number; i++) {
boolean prime = true;
for (int j = 2; j < i; j++) {
if (i % j == 0) {
prime = false;
}
}
if (prime){
return i;
}
}
}
}

You're already there actually. You've just got a mistake in the program-flow.
for (int i = 1; i <= number; i++) {
boolean prime = true;
for (int j = 2; j < i; j++) {
if (i % j == 0) {
prime = false;
}
}
if (prime){
return i; //<-- this return will terminate nextPrim
}
}
Things to fix/improve:
nextPrim would need to return a value within every possible program-branch. This means: consider the case where nextPrim doesn't find any number in the given range and steps out of the loop. Now the program would be stuck without any return-value.
Instead of returning the first prim-number found, you could print that found prim-number and keep the generator running. Nice, easy and solves the hassle with returning anything, since you now can simply declare nextPrim as void. I'd recommend renaming it to printPrims or something like that to make this change clear.
Passing number: You can save a bit of effort by only passing number once to the prim-generator. The simplest solution would be to pass it to nextPrim/printPrims. Now you can remove the instance-variable number and the constructor, which solves the issue with th e signature of the constructor.
1 is not a prim-number to be pedantic. So let's be pedantic and start the outer loop in printPrims with 2, so that 2 will be the first number that is checked for being a prim.
So let's put this into code:
import java.util.Scanner;
public class PrimeGenerator {
public static void main(String args[]) {
Scanner k = new Scanner(System.in);
System.out.println("Enter an integer");
int number = k.nextInt();
PrimeGenerator matt = new PrimeGenerator();
matt.printPrims(number);
}
public void printPrime(int number) {
for (int i = 2; i <= number; i++) {
boolean prime = true;
for (int j = 2; j < i; j++) {
if (i % j == 0) {
prime = false;
}
}
if (prime){
System.out.println(i);
}
}
}
A few general hints:
Work through compiler errors. They tell you precisely where and what errors occur within your code.
Think about the flow of your program before even implementing it.
Break the task down into smaller tasks and implement these one after another. As an example: for this problem first print out all numbers in the range 2, number. Afterwards go a step further and add functionality to filter out prim-numbers. Now you've got two components that you can easily test independent of each other.

You were nearly there but your nextPrimes function was terminating prematurely when you returned i, try something like this:
import java.util.Scanner;
class Main {
public static void main(String[] args) {
Scanner k = new Scanner(System.in);
System.out.print("Enter an integer:");
int number = k.nextInt();
printPrimesUptoN(number);
}
public static void printPrimesUptoN(int n){
for(int i=2;i<n;i++){
boolean isPrime = true;
for(int j=2;j<i;j++){
if(i % j == 0){
isPrime = false;
break;
}
}
if(isPrime)
System.out.println(i);
}
}
}
Try it here!

There's a few issues with your code, maybe we can fix it together. First things first, you're missing a return statement in nextPrime and there's no empty default constructor PrimeGenerator() because you created a single-arg constructor. Try this:
public class PrimeGenerator {
public static void main(String args[]) {
Scanner k = new Scanner(System.in);
System.out.println("Enter an integer");
int number = k.nextInt();
// you probably want to pass your maximum value to the constructor
PrimeGenerator matt = new PrimeGenerator(number);
// without a loop of some sort this will only print a single prime number, e.g.
// Enter a Number:
// 20
// 2
System.out.println(matt.nextPrime(number));
}
private int number;
public PrimeGenerator(int n) {
this.number = n;
}
// you're using the argument as upper boundary for your prime detection while not increasing your lower boundary
// also you're checking if i is a prime here which you always start at 1. this should always return the same value because once you find a prime number you return
// you should consider using an algorithm like Sieve of Eratosthenes (or advanced verions thereof) to determine if a given number is prime
public int nextPrime(int number) {
for (int i = 1; i <= number; i++) {
boolean prime = true;
for (int j = 2; j < i; j++) {
if (i % j == 0) {
prime = false;
}
}
if (prime){
return i;
}
}
// you need to return something at the end of this method or throw an exception
throw new IllegalStateException("no more prime numbers available!");
}
}
This is not a solution which will yield the results you're expecting but it will compile at least. From that point you can move forward and fix the algorithmic issues.

Your original error was in.
PrimeGenerator matt = new PrimeGenerator();
Error:
PrimeGenerator.java:7: error: constructor PrimeGenerator in class PrimeGenerator cannot be applied to given types;
PrimeGenerator matt = new PrimeGenerator();
^
required: int
found: no arguments
reason: actual and formal argument lists differ in length
1 error
Notice that you had a method with the same name as you class, but I don't think you were using it as a constructor and if you were it took an int which you didn't give it. Your method on line 13 was:
public PrimeGenerator(int n) {
number = n;
}
Try doing
new PrimeGenerator().nextPrime(number);
Instead
import java.util.Scanner;
public class PrimeGenerator {
public static void main(String args[]) {
Scanner k = new Scanner(System.in);
System.out.println("Enter an integer");
int number = k.nextInt();
new PrimeGenerator().nextPrime(number);
}
public void nextPrime(int number) {
for (int i = 1; i <= number; i++) {
boolean prime = true;
for (int j = 2; j < i; j++) {
if (i % j == 0) {
prime = false;
}
}
if (prime){
System.out.println(i);
}
}
}
}
Optionally, to use your original constructor, you could separate this out.
import java.util.Scanner;
public class PrimeGenerator {
private int number;
public static void main(String args[]) {
Scanner k = new Scanner(System.in);
System.out.println("Enter an integer");
int number = k.nextInt();
// Initialize with a number.
PrimeGenerator pg = new PrimeGenerator(number);
pg.printPrimes();
}
// This is the constructer you were misusing.
public PrimeGenerator(int n) {
number = n;
}
public void printPrimes() {
// Actually use your private number variable.
for (int i = 1; i <= number; i++) {
boolean prime = true;
for (int j = 2; j < i; j++) {
if (i % j == 0) {
prime = false;
}
}
if (prime){
System.out.println(i);
}
}
}
}

There are several issues:
Are you using an IDE? If so, which is it, else Why not?. If you consider yourself a beginner, be advised to used one, like Eclipse IDE or NetBeans IDE.
Regarding to your algorithm, at first glance, there is a compilation and a logic problems.
Compilation problem is when the execution of your class not work. In this case the method nextPrime(int number) MUST return an int. Although you put return clause, the method is not returning any value when prime is false. You could easily spot this problem with an IDE.
Another compilation problem, the sentinel or variable i is INSIDE the for loop, and you are printing it's value OUTSIDE of it. Once again, the IDE would help you with this problem.
Logic problem is, you are returning a value if and only if the flag prime is true, so, when calling the method (supposing is working) you only obtain ONE value when you call System.out.println(matt.nextPrime(number));
The correct implementation would have these considerations:
The method doesn't return, i.e public void nextPrime(number) { ..., which means you don't need print in the main method, just call it.
The method actually prints the numbers.
In this way, you can check if i % j is different of zero, then you don't need to process more dividers, breaking the for loop.
Print it.
That's it.
public static void main(String [] args) {
.
.
.
PrimeGenerator matt = new PrimeGenerator();
matt.nextPrime(number);
}
public void printPrime(int number) {
boolean prime = true;
for (int i = 2; i <= number; i++) {
prime = true;
for (int j = 2; j < i; j++) {
if (i % j != 0) {
prime = false;
break;
}
}
if (prime) {
System.out.println(i);
}
}
}

Random Prime Methods by calling another method

I am having a difficult time trying to figure out this program. I have made a method to find out if the number entered is a prime number but now I have to write a method that that takes a positive int num as it's parameter and returns a random prime number in the range of [0, num-1]. I'm not sure if I am using the isPrime method correctly. Also I have to test this method by calling it in the main but I'm not sure as how to do that as well. Here is my code:
public static boolean isPrime(int num)
{
for(int i = 2; i<= num-1;i++)
{
if (num % i == 0)
{
return false;
}
}
return true;
}
public static int randomPrime(int num)
{
Random r = new Random();
int x = r.nextInt(num);
for( int i = 0; i <= x; i++)
{
if(!isPrime(x))
{
num = x;
}
}
return x;
}

This is not an efficient way to do it. I would recommend you to construct an Eratosthene sieve and then select a random item of it.
Here is how I would do it :
public static Random rd = new Random(System.currentTimeMillis());
public static int randomPrime(int num) {
Boolean[] sieve = new Boolean[num+1];
sieve[0] = true;
sieve[1] = true;
for (int i=2 ; i<num.length ; i++)
if (!sieve[i])
for (int j = 2*i ; j<num.length ; j += i)
sieve[j] = true;
List<Integer> primes = new LinkedList<>();
for (int i=0 ; i<sieve.length ; i++)
if (!sieve[i]) primes.add(i);
return primes.isEmpty() ? -1 : primes.get(rd.nextInt(primes.size());
}

If you are checking to see if x is prime in your randomPrime method, why are you calling isPrime on 0 <= i <= x? You are checking if any number between (inclusive) 0 and x is prime. That is not what you want.
I think this should do the trick for you without a drastic change of algorithm. Note: This code may take an exceedingly long time to terminate based on your random number generator and seed value and is not actually guaranteed to terminate. (Consider the case where your random number generator only produces composite numbers.)
public static int randomPrime(int num)
{
Random r = new Random();
int x = r.nextInt(num);
while(!isPrime(x))
{
x = r.nextInt(num);
}
return x;
}
Update:
Test it with something like this:
public static void main(String[] args)
{
int N = 42;
int x = randomPrime(N);
System.out.println(x);
}

Can find all the prime numbers but doesn't include "2"?

It now can find all the prime numbers in the input range, but it can't find number 2, the smallest prime number.
for(int number=2;number<range;number++){
for(int testDivide=2;testDivide<Math.sqrt(number);testDivide++){
if(number%testDivide!=0) {
System.out.println(number);
}
break;
}
For range 10 it prints:
5
7
9
but no 2.

The reason your code is not producing correct results (missing 2 and 3; including 9) is that your primality test logic is backwards. A number is prime if the inner loop completes without finding any even divisors; instead you are printing the number if you find any non-divisor.
Try this instead:
for( int number = 2; number < range; number++) {
boolean divisible = false;
int limit = (int) Math.sqrt(number);
for (int testDivide = 2; !divisible && testDivide <= limit; testDivide++) {
divisible = number % testDivide == 0;
}
if (!divisible) {
System.out.println(number);
}
}
Note that a much more efficient way to generate all primes in a range is the Sieve of Eratosthenes.

check the code here:
package core;
public class Test2 {
public static void main(String[] args) {
int cnt = 0;
for (int i = 2;; i++) {
if (Priem(i)) {
cnt++;
System.out.println(i);
if (cnt == 200)
break;
}
}
}
public static boolean Priem(int n) {
for (int i = 2; i < n; i++) {
if (n % i == 0) {
return false;
}
}
return true;
}
}

Note that one of the best ways to generate a list of prime numbers is the "Sieve of Erwhatshisface":
Create a list of consecutive integers starting with 2, up to the max number you'd like to search. Take the first non-zero number in the list (2) and repeatedly step 2 from that location, zeroing out every 2nd list element.
Next take the second non-zero number (3) and repeatedly step 3 from that location, zeroing. Continue with each non-zero value in the list, until you've processed all of them (or at least halfway through, at which point you'll be stepping beyond the end of the list).
The non-zero numbers remaining are all primes.

Reposting code here as not fit in comments:
public static void main(String[] args) {
int cnt = 0;
for (int i = 2;; i++) {
if(i==2){
System.out.println(i);
continue;
}
if (Priem(i)) {
cnt++;
System.out.println(i);
if (cnt == 200)
break;
}
}
}
public static boolean Priem(int n) {
for (int i = 2; i <Math.sqrt(n)+1; i++) {
if (n % i == 0) {
return false;
}
}
return true;
}

I think you have your for loop a little mixed up.
for(int testDivide=2;testDivide<Math.sqrt(number);testDivide++){
}
Not sure why you are stopping when testDivide is equal to sqrt(number), you should stop when testDivide is greater than.
Also inside your inner for loop isn't correct either:
if(number%testDivide!=0) {
System.out.println(number);
}
break;
Basically all this will do is check to see of the number is divisible by 2 and then break. You only need to break when you find a number which cleanly divides (number%testDivide==0). Maybe keep a boolean which you set to true when you break and only print after the inner for loop finishes if that boolean is false.
Something along the lines:
for (int number=2; number<range; number++){
boolean found = false;
int limit = (int)Math.sqrt(number);
for (int testDivide=2; testDivide<=limit; testDivide++){
if(number%testDivide==0) {
found = true;
break;
}
}
if (!found) System.out.println(number);
}

In your code when number is 2, sqrt(2) is 1.41 and control doesn't go into the loop. I didn't get the logic behind iterating upto sqrt(number). Try this code
public class Test {
public static void main(String[] args) {
int range = 500; //I assume
for (int i = 2; i< range; i++) {
if (isPrime(i)) {
System.out.println(i);
}
}
}
public static boolean isPrime(int number) {
for (int i = 2; i <= number/2; i++) {
if (number % i == 0) {
return false;
}
if(i % 2 == 1) {
i++; //If not divided by 2 then
// need not to check for any even number
// Essentially incrementing i twice hereafter
}
}
return true;
}
}

Wrong Answer for Project Euler 50

I am attempting Problem 50 of project Euler.
The prime 41, can be written as the sum of six consecutive primes:
41 = 2 + 3 + 5 + 7 + 11 + 13 This is the longest sum of consecutive
primes that adds to a prime below one-hundred. The longest sum of
consecutive primes below one-thousand that adds to a prime, contains
21 terms, and is equal to 953. Which prime, below one-million, can be
written as the sum of the most consecutive primes?
Here is my code:
public class consPrime
{
static int checker(int ar[],int num,int index) //returns no.of consecutive
{ //primes for the given num
while(true)
{
int temp=num;
for(int i=index;i>=0;i--)
{
temp=temp-ar[i];
if(temp==0)
{
return (index-i+1);
}
}
index--;
if(index==0)
return 0;
}
}
public static void main(String args[])
{
int n=100000;
int ar[]=new int[n];
int total=0;int flag;
for(int i=2;i<1000000;i++) //Generates an array of primes below 1 million
{
flag=1;
for(int j=2;j<=Math.sqrt(i);j++)
{
if(i%j==0)
{
flag=0;
break;
}
}
if(flag==1)
{
ar[total]=i;
total++;
}
}
int m=0;
int Big=0;
for(int i=total;i>=0;i--) //Prints the current answer with no.of prime
{
m=checker(ar,ar[i],i-1);
if(Big<=m)
{Big=m;
System.out.println(ar[i]+" "+Big);
}
}
}
}
Basically it just creates a vector of all primes up to 1000000 and then loops through them finding the right answer. The answer is 997651 and the count is supposed to be 543 but my program outputs 990707 and 75175 respectively. What might be wrong?

Several big problems:
Some minor problem first: learn to proper indent your code, learn to use proper naming convention. In Java, variable names uses camelCasing while type name uses PascalCasing.
Lots of problems in your logics: you loop thru the prime number array, until you hit zero or until looped thru all numbers in the array. However, please be awared that, there is underflow/overflow for integer. It is possible that the "temp" keeps on deducts and become negative and become positive and so-on-and-so-forth and hit zero. However that's not the correct answer
You only tried to find the consecutive numbers that ends at index - 1. For example, to check for prime number at index 10, you are finding consecutive primes from index 9 backwards. However consecutive prime sum up to your target number rarely (in fact almost never, except for 5) contains the "previous" prime number. The whole logic is simply wrong.
Not to mention the incorrect parameters you passed for checker, which is mentioned by comment of user #pm-77-1

Here is another approach that takes 43 ms.
It is based on the following approach:
1) The primes <= 1000000 are generated using a sieve
2) It iterates in O(n2) through all numbers and it counts the consecutive primes. The first loop changes the first element of the sequence, the second one takes the elements starting from that position and adds them to a sum. If the sum is prime and it consists of the biggest number of primes, than it is kept in a variable.
import java.util.ArrayList;
import java.util.List;
public class P50 {
private final static int N = 1_000_000;
public static void main(String[] args) {
boolean primes[] = generatePrimes(N);
List<Integer> primeIntegers = new ArrayList<Integer>();
for (int i = 0; i < primes.length; i++) {
if (primes[i]) {
primeIntegers.add(i);
}
}
int count = 0;
int sum = 0;
int finalSum = 0;
int finalCount = 0;
int totalPrimes = primeIntegers.size();
for (int start = 0; start < totalPrimes; start++) {
sum = 0;
count = 0;
for (int current = start; current < totalPrimes; current++) {
int actual = primeIntegers.get(current);
sum += actual;
if ( sum >= N ) {
break;
}
if ( primes[sum] ) {
if ( count > finalCount ) {
finalCount = count;
finalSum = sum;
}
}
count++;
}
}
System.out.println(finalSum);
}
private static boolean[] generatePrimes(int n) {
boolean primes[] = new boolean[n];
for (int i = 0; i < n; i++) {
primes[i] = true;
}
primes[0] = false;
primes[1] = false;
// i = step
for (int i = 2; i * i < n; i++) {
if (primes[i]) {
for (int j = i * i; j < n; j += i) {
primes[j] = false;
}
}
}
return primes;
}
}