# 1051 is the 21st Centered Pentagonal Number

1051 is the 21st centered pentagonal number. It is exactly 100 more than the previous centered pentagonal number because there are exactly 100 little blue squares on the outside-most pentagon in the graphic below. Can you see the five triangles surrounding the center square? Each of them has the same number of tiny squares and indicates that 1051 is 1 more than five times the 20th triangular number:
1 + 5(20)(21)/2 = 1 + 50(21) = 1051

1049 and 1051 are twin primes.

• 1051 is a prime number.
• Prime factorization: 1051 is prime.
• The exponent of prime number 1051 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1051 has exactly 2 factors.
• Factors of 1051: 1, 1051
• Factor pairs: 1051 = 1 × 1051
• 1051 has no square factors that allow its square root to be simplified. √1051 ≈ 32.41913

How do we know that 1051 is a prime number? If 1051 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1051 ≈ 32.4. Since 1051 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1051 is a prime number. 1051 is a palindrome when it is written in three other bases:
It’s 737 in BASE 12 because 7(144) + 3(12) + 7(1) = 1051,
1H1 in BASE 25 (H is 17 base 10) because 25² +17(25) + 1 = 1051, and
151 in BASE 30 because 30² + 5(30) + 1 = 1051

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