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Sum a series n^n for values 1 through n with no overflow? Only last digits of answer needed

I want to write a Java program that sums all the integers n^n from 1 through n. I only need the last 10 digits of this number, but the values given for n exceed 800.
I have already written a basic java program to calculate this, and it works fine for n < 16. But it obviously doesn't deal with such large numbers. I am wondering if there is a way to just gather the last 10 digits of a number that would normally overflow a long, and if so, what that method or technique might be.
I have no code to show, just because the code I wrote already is exactly what you'd expect. A for loop that runs i*i while i<=n and a counter that sums each iteration with the one before. It works. I just don't know how to approach the problem for bigger numbers, and need guidance.
Around n=16, the number overflows a long, and returns negative values. Will BigInteger help with this, or is that still too small a data type? Or could someone point me towards a technique for gathering the last 10 digits of a massive number? I could store it in an array and then sum them up if I could just get that far.
Anyhow, I don't expect a finished piece of code, but maybe some suggestions as to how I could look at this problem anew? Some techniques my n00b self is missing?
Thank you!

sums all the integers n^n from 1 through n. I only need the last 10 digits of this number
If you only need last 10 digits, that means you need sum % 10¹⁰.
The sum is 1¹ + 2² + 3³ + ... nⁿ.
According to equivalences rules:
(a + b) % n = [(a % n) + (b % n)] % n
So you need to calculate iⁱ % 10¹⁰, for i=1 to n, sum them, and perform a last modulus on that sum.
According to the modular exponentiation article on Wikipedia, there are efficient ways to calculate aⁱ % m on a computer. You should read the article.
However, as the article also says:
Java's java.math.BigInteger class has a modPow() method to perform modular exponentiation
Combining all that to an efficient implementation in Java that doesn't use excessive amounts of memory:
static BigInteger calc(int n) {
final BigInteger m = BigInteger.valueOf(10_000_000_000L);
BigInteger sum = BigInteger.ZERO;
for (int i = 1; i <= n; i++) {
BigInteger bi = BigInteger.valueOf(i);
sum = sum.add(bi.modPow(bi, m));
}
return sum.mod(m);
}
Or the same using streams:
static BigInteger calc(int n) {
final BigInteger m = BigInteger.valueOf(10).pow(10);
return IntStream.rangeClosed(1, n).mapToObj(BigInteger::valueOf).map(i -> i.modPow(i, m))
.reduce(BigInteger.ZERO, BigInteger::add).mod(m);
}
Test
System.out.println(calc(800)); // prints: 2831493860

BigInteger would be suitable to work with these kinds of numbers. It's quite frankly what it's designed for.
Do note that instances of BigInteger are immutable and any operations you do on one will give you back a new BigInteger instance. You're going to want to store some of your results in variables.

How does index*int in a for loop end up with zero as result?

We tried just for fun to create a for loop like below. We assumed that the number we get would be very high but we got 0. Why is it 0 and not something big?
We even tried it with a long because we thought it might be bigger than a int.
Thanks in advance.
private static void calculate() {
int currentSolution = 1;
for (int i = 1; i < 100; i++) {
currentSolution *= i;
}
System.out.println(currentSolution);
}

Your int is wrapping round to -2147483648 when it reaches +2147483647.
By an amazing coincidence1, a multiplication by zero is introduced into your product.
See for yourself: write
if (currentSolution == 0){
// What is the value of i?
}
You'll need a BigInteger to evaluate 100!.
1 Really it's not that amazing: it's just that 100! has 232 as a factor.

Your for loop calculates the factorial of 100, i.e. 1 * 2 * 3 * ... * 99 * 100, also written as 100! which equals 9.332621544×10157
The range of int is -2,147,483,648 to 2,147,483,647 and the range of long is -9,223,372,036,854,775,808 to 9,223,372,036,854,775,807, so with a number that is 10157 you've hit multiplication overflow long before you even get close to the end of the for loop.
The overflow eventually results in all bits eventually zeroing out, thus producing the result 0.

Your datatypes are too small, even if you use long, so you cannot do this calculation with basic types. So you get an overflow, where the values become negative, and at some point 0. Therefore, the result is still zero after every loop.
You can see that by printing the currentSolution in the loop.
In order to get the correct solution, try using BigInteger:
public static void calculate() {
BigInteger currentSolution = BigInteger.valueOf(1);
for (int i = 1; i < 100; i++) {
currentSolution = currentSolution.multiply(BigInteger.valueOf(i));
}
System.out.println(currentSolution);
}
This outputs the correct solution: 933262154439441526816992388562667004907159682643816214685929638952175999932299156089414639761565182862536979208272237582511852109168640000000000000000000000

You re basically calculating 100!, which is approximately 9.3e+157, whereas a long maximum value is 2^64 -1 so approximately 1.8e+19.
It cant fit, thats why you have 0.

Luhn checksum validation in Java

I have to replicate the luhn algorithm in Java, the problem I face is how to implement this in an efficient and elegant way (not a requirement but that is what I want).
The luhn-algorithm works like this:
You take a number, let's say 56789
loop over the next steps till there are no digits left
You pick the left-most digit and add it to the total sum. sum = 5
You discard this digit and go the next. number = 6789
You double this digit, if it's more than one digit you take apart this number and add them separately to the sum. 2*6 = 12, so sum = 5 + 1 = 6 and then sum = 6 + 2 = 8.
Addition restrictions
For this particular problem I was required to read all digits one at a time and do computations on each of them separately before moving on. I also assume that all numbers are positive.
The problems I face and the questions I have
As said before I try to solve this in an elegant and efficient way. That's why I don't want to invoke the toString() method on the number to access all individual digits which require a lot of converting. I also can't use the modulo kind of way because of the restriction above that states once I read a number I should also do computations on it right away. I could only use modulo if I knew in advance the length of the String, but that feels like I first have to count all digits one-for-once which thus is against the restriction. Now I can only think of one way to do this, but this would also require a lot of computations and only ever cares about the first digit*:
int firstDigit(int x) {
while (x > 9) {
x /= 10;
}
return x;
}
Found here: https://stackoverflow.com/a/2968068/3972558
*However, when I think about it, this is basically a different and weird way to make use of the length property of a number by dividing it as often till there is one digit left.
So basically I am stuck now and I think I must use the length property of a number which it does not really have, so I should find it by hand. Is there a good way to do this? Now I am thinking that I should use modulo in combination with the length of a number.
So that I know if the total number of digits is uneven or even and then I can do computations from right to left. Just for fun I think I could use this for efficiency to get the length of a number: https://stackoverflow.com/a/1308407/3972558
This question appeared in the book Think like a programmer.

You can optimise it by unrolling the loop once (or as many times are you like) This will be close to twice as fast for large numbers, however make small numbers slower. If you have an idea of the typical range of numbers you will have you can determine how much to unroll this loop.
int firstDigit(int x) {
while (x > 99)
x /= 100;
if (x > 9)
x /= 10;
return x;
}

use org.apache.commons.validator.routines.checkdigit.LuhnCheckDigit . isValid()
Maven Dependency:
<dependency>
<groupId>commons-validator</groupId>
<artifactId>commons-validator</artifactId>
<version>1.4.0</version>
</dependency>

Normally you would process the numbers from right to left using divide by 10 to shift the digits and modulo 10 to extract the last one. You can still use this technique when processing the numbers from left to right. Just use divide by 1000000000 to extract the first number and multiply by 10 to shift it left:
0000056789
0000567890
0005678900
0056789000
0567890000
5678900000
6789000000
7890000000
8900000000
9000000000
Some of those numbers exceed maximum value of int. If you have to support full range of input, you will have to store the number as long:
static int checksum(int x) {
long n = x;
int sum = 0;
while (n != 0) {
long d = 1000000000l;
int digit = (int) (n / d);
n %= d;
n *= 10l;
// add digit to sum
}
return sum;
}

As I understand, you will eventually need to read every digit, so what is wrong with convert initial number to string (and therefore char[]) and then you can easily implement the algorithm iterating that char array.
JDK implementation of Integer.toString is rather optimized so that you would need to implement your own optimalizations, e.g. it uses different lookup tables for optimized conversion, convert two chars at once etc.
final static int [] sizeTable = { 9, 99, 999, 9999, 99999, 999999, 9999999,
99999999, 999999999, Integer.MAX_VALUE };
// Requires positive x
static int stringSize(int x) {
for (int i=0; ; i++)
if (x <= sizeTable[i])
return i+1;
}
This was just an example but feel free to check complete implementation :)

I would first convert the number to a kind of BCD (binary coded decimal). I'm not sure to be able to find a better optimisation than the JDK Integer.toString() conversion method but as you said you did not want to use it :
List<Byte> bcd(int i) {
List<Byte> l = new ArrayList<Byte>(10); // max size for an integer to avoid reallocations
if (i == 0) {
l.add((byte) i);
}
else {
while (i != 0) {
l.add((byte) (i % 10));
i = i / 10;
}
}
return l;
}
It is more or less what you proposed to get first digit, but now you have all you digits in one single pass and can use them for your algorythm.
I proposed to use byte because it is enough, but as java always convert to int to do computations, it might be more efficient to directly use a List<Integer> even if it really wastes memory.

Which data type or data structure to choose to calculate factorial of 100?

I thought of writing a program to evaluate factorial of a given integer.
Following basics I wrote the below code in java :
long fact(int num){
if(num == 1)
return 1;
else
return num*fact(num-1);
}
But then I realized that for many integer input the result may not be what is desired and hence for testing directly gave input as 100.
My doubt was true as Result I got was "0"(cause result might be out of range of long).
So,I am just curious and eager to know as how may I make my program work for inputs<=150.
I would appreciate any valid solution in C programming language or Java.

BigInteger is your class. It can store integers of seemingly any size.
static BigInteger fact(BigInteger num) {
if (num.equals(BigInteger.ONE))
return BigInteger.ONE;
else
return num.multiply(fact(num.subtract(BigInteger.ONE)));
}

If you're not after a naive approach of factorial computation, you should do some research into the problem. Here's a good overview of some algorithms for computing factorials: http://www.luschny.de/math/factorial/conclusions.html
But like the other answers suggest, your current problem is that you need to use a large number implementation (e.g. BigInt) instead of fixed size integers.

In C Language, you can use array to store factorial of large number.
my reference: Calculate the factorial of an arbitrarily large number, showing all the digits. it very helpful post.
I made small changes in code to convert into C.
int max = 5000;
void factorial(int arr[], int n){//factorial in array
if (!n) return;
int carry = 0;
int i=max-1;
for (i=max-1; i>=0; --i){
arr[i] = (arr[i] * n) + carry;
carry = arr[i]/10;
arr[i] %= 10;
}
factorial(arr,n-1);
}
void display(int arr[]){// to print array
int ctr = 0;
int i=0;
for (i=0; i<max; i++){
if (!ctr && arr[i])
ctr = 1;
if(ctr)
printf("%d", arr[i]);
}
}
int main(){
int *arr = calloc(max, sizeof(int));
arr[max-1] = 1;
int num = 100;
printf("factorial of %d is: ",num);
factorial(arr,num);
display(arr);
free(arr);
return 0;
}
And its working for 100! see: here Codepad
I would like to give you links of two more useful posts.
1) How to handle arbitrarily large integers suggests GPU MP
2) C++ program to calculate large factorials

In java you have the BigInteger that can store arbitrary big integers. Unfortunately there is no equivelent in C. You either have to use a third-party library or to implement big integers on your own. Typical approach for this is to have a dynammically-allocated array that stores each of the digits of the given number in some numeric system(usually base more than 10 is chosen so that you reduce the total number of digits you need).

A decimal (base 10) digit takes about 3.3 bits (exactly: log(10)/log(2)). 100! is something like 158 digits long, so you need 158 * 3.3 = 520 bits.
There is certainly no built in type in C that will do this. You need some form of special library if you want every digit in the factorial calculation to be "present".
Using double would give you an approximate result (this assumes that double is a 64-bit floating point value that is IEEE-754 compatible, or with similar range - the IEEE-754 double format will give about 16 decimal digits (52 bits of precision, divided by the log(10)/log(2) like above). I believe there are more than 16 digits in this value, so you won't get an exact value, but it will calculate some number that is within a 10 or more digits.

How to test if an array contains a pair of numbers whose product is odd?

How can I write a function that takes an array of integers and returns true if their exists a pair of numbers whose product is odd?
What are the properties of odd integers? And of course, how do you write this function in Java? Also, maybe a short explanation of how you went about formulating an algorithm for the actual implementation.
Yes, this is a function out of a textbook. No, this is not homework—I'm just trying to learn, so please no "do your own homework comments."

An odd number is not evenly divisible by two. All you need to know is are there two odd numbers in the set. Just check to see if each number mod 2 is non-zero. If so it is odd. If you find two odd numbers then you can multiply those and get another odd number.
Note: an odd number multiplied by an even number is always even.

The product of two integers will be odd only if both integers are odd. So, to solve this problem, just scan the array once and see if there are two (or more) odd integers.
EDIT: As others have mentioned, you check to see if a number is odd by using the modulus (%) operator. If N % 2 == 0, then the number is even.

Properties worth thinking about:
Odd numbers are not divisible by 2
Any number multiplied by an even number is even
So you can re-state the question as:
Does the array contain at least two integers that are not divisible by 2?
Which should make things easier.

You can test for evenness (or oddness) by using the modulus.
i % 2 = 0 if i is even; test for that and you can find out if a number is even/odd

A brute force algorithm:
public static boolean hasAtLeastTwoOdds(int[] args) {
int[] target = args; // make defensive copy
int oddsFound;
int numberOddsSought = 2;
for (int i = 0; i < target.length; i++) {
if (target[i] % 2 != 0) {
if (oddsFound== numberOddsSought) {
return true;
}
oddsFound++;
}
}
return false;
}

Thank you for your answers and comments.
I now understand well how to test whether an integer is odd. For example, this method is a neat way of doing this test without using multiplication, modulus, or division operators:
protected boolean isOdd(int i) {
return ( (i&1) == 1);
}
With your help, I now realize that the problem is much simpler than I had expected. Here is the rest of my implementation in Java. Comments and criticism are welcome.
protected boolean isOddProduct(int[] arr) {
int oddCount = 0;
if (arr.length < 2)
throw new IllegalArgumentException();
for (int i = 0; i <= arr.length-1; i++) {
if (isOdd(arr[i]))
oddCount++;
}
return oddCount > 1;
}
I wonder if there exists any other ways to perform this test without using *, % or / operators? Maybe I'll ask this question in a new thread.