Integers are a set of whole numbers that include all positive and negative whole numbers, as well as zero. In mathematical terms, the set of integers is usually denoted by ℤ.
Examples of Integers:
- Positive Integers: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, . . .
- Negative Integers: −1, −2, −3, -4, -5, -6, -7, -8, -9, -10, -11, -12, . . .
- Zero: 0
Some interesting facts about Integers are:
1. There is no smallest or largest integer, meaning the set of integers is infinite.
2. Integers are countably infinite, unlike real numbers.
3. Negative numbers were not accepted on the number line until the 19th century.
4. Various famous sequences, such as the Fibonacci sequence, Lucas numbers, and Catalan numbers, are comprised entirely of integers.
5. For any integer n, the last digit of n5 is always the same as the last digit of n. For example:
- 25 = 32 (last digit 2)
- 35 = 243 (last digit 3)
- 45 = 1024 (last digit 4)
- 55 = 3125 (last digit 5)
- 65 = 7776 (last digit 6)
- 75 = 16807 (last digit of 7)
- 85 = 32768 (last digit of 8)
- 95 = 59049 (last digit of 9)
6. For any prime number p, the expression (p−1)! + 1 is also divisible by p.
- For example, For p = 7: (7 − 1)! + 1 = 6! + 1 = (6 × 5 × 4 × 3 × 2 × 1) + 1 = 720 + 1 = 721 (which is divisible by 7).
7. Every natural number can be expressed as the sum of four perfect squares. This is known as Lagrange's Four-Square Theorem.
- For example, For the number 7: 7 = 4 + 1 + 1 + 1 (4 = 22, 1 = 12, 1 = 12, 1 = 12).
8. The number 13223140496 is the smallest integer that, when concatenated with itself, forms a perfect square, For example:
- When 13223140496 is concatenated with itself, it becomes: 363636363642 = 1322314049613223140496 (perfect square).
9. Belphegor's Prime is a unique palindromic prime number, written as: 1000000000000066600000000000001
10. The number 2592 is known as the printer's error because it can be expressed as: 2592 = 25 × 92(It is the only known number of this kind).
11. The number 999999937 is prime and is the largest prime number less than 1 billion.
12. 1026753849 is the smallest perfect square that contains each digit from 0 to 9 exactly once, represented as 320432.