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 twodigit power
numbers and 28 threedigit 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

%

19

4

44.44

19

4

44.44

1099

8

8.89

199

12

12.12

100999

28

3.11

1999

40

4.4

10009999

84

.93

19999

124

1.28

1000099999

242

.269

199999

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 n^{2}

n

Last digits of n^{2}


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_{k}d_{k1}...d_{4}d_{3}d_{2}d_{1}
be a kdigit number.
Step 1: Take the square of the last digit i.e., d_{1}. Last
digit of it is the last digit of the square.
Keep the carryover, if any.
Step 2: Multiply d_{2} with d_{1} 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 d_{2}. Multiply d_{3}
with d_{1} 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 d_{4} with d_{1}, d_{3}
with d_{2 }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 2k1 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., 10, 41, 94, 169, 2516, ... In general, the n^{th}
odd number, say, O_{n}=n^{2}(n1)^{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}(n1)^{2}=2^{2}n. Therefore, n= [(n+1)^{2}(n1)^{2}]/2^{2}.
Prime Numbers and Square Numbers
For any given number, we know that the largest nontrivial 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 kdigit powers numbers always an
even number? Is there a better connection between square numbers and prime
numbers? What are the properties of the last kdigits 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.