Square Numbers

Natural numbers are numbers that are used for counting. Among the natural numbers there are some numbers called power numbers. They are nothing but natural numbers which can be written as the product of a number with itself two or more times. Number 1 is multiplied with itself any number of times would give only 1. Therefore, 1 is a trivial power number. Power number is also understood as a number n with the equation x^k=n that has the integer solution of for some positive integer k>1.

Number 2 is the number that follows number 1 among the natural numbers. We can easily see that 2 cannot be written as product of identical numbers. Next natural number 3 also has the similar property. Both 2 and 3 are also the first two prime numbers. A prime number is a number that cannot be written as the product two distinct numbers other than 1 and itself.

Power Numbers

But the next natural number 4 is a power number. It is the product of 2 with itself. In connection with the concept of prime numbers, we can say that a power number is a composite number with identical factors. By inspection we can see that among the single digit numbers, only 1, 4, 8 and 9 are power numbers. Power numbers are very rare among the natural numbers. There are only 8 two-digit power numbers and 28 three-digit power numbers.  Among the entire 9000 four digit numbers there are only 84 power numbers. Strange though, the percentage of power numbers below is very meagre for large values of n. For example, there are only .366% of power numbers among all numbers with at most five digits.

Range	Number	%	Range	Number	%
1-9	4	44.44	1-9	4	44.44
10-99	8	8.89	1-99	12	12.12
100-999	28	3.11	1-999	40	4.4
1000-9999	84	.93	1-9999	124	1.28
10000-99999	242	.269	1-99999	366	.366

Square Numbers

Among the single digit power numbers 1, 4 and 9 are called square numbers. A square number is the product of an integer with itself.  The term ‘square’ is attributed to these numbers because of the fact that it is the area of a square with an integer side length.  Number 8 is a cube number because it is the product of 2 three times.

Square Number and the last digit

Let us see the some of the properties of square numbers. First eleven square numbers are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121 ... These square numbers help us in observing a salient feature of square numbers. That is, a square number will never end in 2, 3, 7 or 8. This is a fundamental property of the square numbers.

Among the first 10 square numbers, it can be observed that square of both 1 and 9 end in 1. Similarly squares of 2 and 8 end in 4, squares of 3 and 7 end in 9 and that of 4 and 6 end in 6. It is interesting to note that 1+9=2+8=3+7=4+6=10. Hence, we can conclude that square of a number of the form 10k∓n, 1≤n≤4, ends in 1, 4, 9 and 6 respectively whereas the square of a number that ends in 0 or5 will always end in 0 and5, respectively. It is interesting to note that among these end digits 5 and 6 are not square numbers whereas all others are.

Square Numbers and the last two digits

Now, let us look at the properties of the last two digits of a square number. Though 1, 5, 6 and 9 can be the end digits of square numbers, if they are repeated, then they cannot be the end digits of square numbers. That is to say that only 00 and 44 can be the last two digits of square numbers. Among the first 100 numbers there are exactly 22 numbers that can be the end digits of square numbers. They are 00, 01, 04, 09, 16, 21, 24, 25, 29, 36, 41, 44, 49, 56, 61, 64, 69, 76, 81, 84, 89 and 96.

There are some interesting properties for these numbers. The numbers 00, 16, 44 and 69 in the reverse order also can be the end digits of square numbers. Sum of two single digit numbers can have any value from 0 to 18. But, the sum of the last two digits of square numbers can never be 2, 14, 16 and 18. Hence, similar to that of single digit ends of square numbers, we have a result.

Given below are the types of numbers whose squares have particular end digits. Here k is any nonnegative integer.

n	Last digits of n2		n	Last digits of n2
10k	00		10k	00
10k∓5	25		10k∓5	25
50k∓1	01		50k∓1	01
50k∓2	04		50k∓2	04
50k∓3	09		50k∓3	09
50k∓4	16		50k∓4	16
50k∓11	21		50k∓6	36
50k∓18	24		50k∓7	49
50k∓23	29		50k∓8	64
50k∓6	36		50k∓9	81
50k∓21	41		50k∓11	21
50k∓12	44		50k∓12	44
50k∓7	49		50k∓13	69
50k∓16	56		50k∓14	96
50k∓19	61		50k∓16	56
50k∓8	64		50k∓17	89
50k∓13	69		50k∓18	24
50k∓24	76		50k∓19	61
50k∓9	81		50k∓21	41
50k∓22	84		50k∓22	84
50k∓17	89		50k∓23	29
50k∓14	96		50k∓24	76

Algorithm to find the square

There is an Indian short cut to find the square of a number.  Though the algorithm looks tedious and lengthy, we can write the square in a line, if we master the algorithm.

Let d_kd_k-1...d₄d₃d₂d₁ be a k-digit number.

Step 1: Take the square of the last digit i.e., d₁. Last digit of it is the last digit of the square.  Keep the carryover, if any.

Step 2: Multiply d₂ with d₁ and take twice of it and add the carryover from the previous step. The last digit of this number is the last but one digit of the square. Keep the carryover, if any.

Step 3: Take the square of d₂. Multiply d₃ with d₁ and take twice of it. Add these two sums with the carryover from Step 3. The last digit of this number is the e last but two digit of the square. Keep the carryover, if any.

Step 4: Multiply d₄ with d₁, d₃ with d₂ and take twice of both and add the carryover from the previous step. The last digit of this number is the last but three digit of the square. Keep the carryover, if any.

Continuing in this manner, in 2k-1 steps, we get the required square.

Triangular Numbers and Square Numbers

Square numbers have a fascinating link to triangular numbers. A triangular number is the sum of first n consecutive natural numbers. They are 1, 3, 6, 10, 15 ...  It is striking to note that any square number is the sum of two consecutive triangular numbers i.e., 1=1+0, 4=3+1,  9=6+3, 16=10+6, ... This leads us to conclude that twice the sum of first n natural numbers added to n+1 is a square number. i.e., if T_n is the n^th triangular number, then 2T_n+ (n+1) and 2T_n-n are square numbers.

Parity of Numbers and Square Numbers

Alas! We have another result. Difference of two consecutive square numbers is always an odd number.  Therefore, we can have the sequence of odd numbers as the difference of square numbers viz., 1-0,  4-1, 9-4, 16-9, 25-16, ... In general, the n^th odd number, say, O_n=n²-(n-1)².

Similarly, even numbers also can be found from square numbers. The parity of square numbers is same as the parity of natural numbers. Hence, the difference between every pair of alternate square numbers is as an even number. Moreover, they are all multiples of 4. Halving them we get the sequence of even numbers. Quartering them we get the natural numbers. That is to say that for any natural number n, (n+1)²-(n-1)²=2²n.  Therefore, n= [(n+1)²-(n-1)²]/2².

Prime Numbers and Square Numbers

For any given number, we know that the largest non-trivial factor it can have is its square root. Hence, in finding the factors of a number one needs to check up to its square root only. Are square numbers some way helpful to know about the distribution of prime numbers? It is easy to see that for any given n, the number of prime numbers less than n is much larger than the number of square numbers. We have seen earlier square numbers alternates parity. This gives us some clue. Take only the even square numbers. Its immediate number is likely to be a prime number.  That is, the sequence, a_n= (2n)²+1, where n does not end in 1, 4, 6 or 9, has plenty of prime numbers.  Some of the initial members of the sequence are 17, 37, 101, 197, 257, 401, 577, 677, 901, 1157, 1297, 1601,  ... Some remembrances of the sequences of Mersenne Primes and Fermat Primes.

For training in mathematics research, the collection of square numbers is a good area. Following questions are worth searchable for beginners in number theory research. Is the number of k-digit powers numbers always an even number? Is there a better connection between square numbers and prime numbers? What are the properties of the last k-digits of power (square) numbers? Various proofs for such properties would really be a hard test. How far the properties of power numbers help us in solving the integer factorization problem? To sum up, one can escalate one’s mathematical powers by doing research on power numbers and can become a square personality mathematically.