If you only look at a list of triangular numbers or a list of hexagonal numbers, you might miss the relationship that figurate numbers have with each other.
1378 is the 52nd triangular number, and you can use it to find the 53rd triangular number (1431), the 53rd square number, the 53rd pentagonal number, and so forth.
351 is the 26th triangular number, and you can use it to find the 27th triangular number, the 27th square number, the 27th pentagonal number, the 27th hexagonal number (1431), and so forth.
See the relationship in the graphic below:
Should you get excited that 1431 is BOTH a triangular number and a hexagonal number? Not really. It turns out that every hexagonal number is also a triangular number. (But not every triangular number is a hexagonal number.)
Here are a few more facts about the number 1431:
- 1431 is a composite number.
- Prime factorization: 1431 = 3 × 3 × 3 × 53, which can be written 1431 = 3³ × 53.
- 1431 has at least one exponent greater than 1 in its prime factorization so √1431 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1431 = (√9)(√159) = 3√159.
- The exponents in the prime factorization are 3 and 1. Adding one to each exponent and multiplying we get (3 + 1)(1 + 1) = 4 × 2 = 8. Therefore 1431 has exactly 8 factors.
- The factors of 1431 are outlined with their factor pair partners in the graphic below.
1431 is the hypotenuse of a Pythagorean triple:
756-1215-1431 which is 27 times (28-45-53)