The number 35, often referred to as thirty-five, is a natural number that follows 34 and precedes 36. It is a triangular number and the 10th discrete semiprime, meaning it has exactly two distinct prime factors. The number 35 holds several interesting mathematical properties and has applications in various fields. It is the sum of the first five triangular numbers and is a tetrahedral number. The aliquot sum of 35 is 13, making it a member of the unique aliquot sequence {35, 13, 1, 0}. In science, the atomic number of bromine is 35, and the number 35 is used for calculating retirement benefits in the United States, Ireland, Poland, Russia, Trinidad and Tobago, and Uruguay. The minimum age required for presidential candidates to be eligible for elections in those countries is 35. In other fields, 35mm film is the standard film gauge for both analog photography and motion pictures. The number 35 is also significant in the realm of social security, as the 35 highest years of earnings are used to determine retirement benefits in the United States.
Mathematical Properties of 35
Tetrahedral Number
35 is a tetrahedral number, a term that refers to the number of regions formed by connecting the vertices of a regular tetrahedron with its opposite faces. The relationship between triangular numbers and tetrahedral numbers is given by the formula:
T_{n} = n * (n + 1) / 2
where T_{n} is the nth triangular number and n is a positive integer. Since T_{6} = 15 and T_{7} = 21, the 35th triangular number is 15 * 16 / 2 = 120. Therefore, the number 35 is indeed a tetrahedral number.
Sum of Triangular Numbers
35 is also the sum of the first five triangular numbers:
T_{1} = 1
T_{2} = 1 + 2
T_{3} = 1 + 2 + 3
T_{4} = 1 + 2 + 3 + 4
T_{5} = 1 + 2 + 3 + 4 + 5
Adding these terms together yields:
35 = 1 + 2 + 3 + 4 + 5 = 15
Therefore, the number 35 is the sum of the first five triangular numbers.
Discrete Semi Prime
35 is a 10th discrete semiprime, meaning it has exactly two prime factors. The two prime factors of 35 are 5 and 7, which also form its main factor pair (5 × 7). This makes 35 a twin-prime distinct semiprime pair along with the number 33.
Aliquot Sum
The aliquot sum of 35 is 13, within an aliquot sequence of only one composite number (35,13,1,0) to the Prime in the 13-aliquot tree. This indicates that 35 is the second composite number with the aliquot sum 13; the first being the cube 27. The aliquot sum of 35 is the sum of the aliquot sequences up to and including its aliquot root 13.
Triple Cluster of Semiprimes
35 is the last member of the first triple cluster of semiprimes 33, 34, 35. The second such triple distinct semiprime cluster is 85, 86, and 87. This clustering of prime numbers is notable and may have implications for prime number distribution theories.
combination of seven things taken three at a time
The number 35 is a centered cube number, a centered tetrahedral number, a pentagonal number, and a pentatope number, making it a highlycototient. This property means that there are more solutions to the equation x – 1 = x^n for n > 1 and x = 35 than for any other integer below 35 except 1.
In Science and Other Fields
Atomic Number of Bromine
The atomic number of bromine is 35, which is essential for the synthesis of numerous chemical compounds. bromide ions are commonly found in食盐 and many seafoods, while bromine is used as an industrial disinfectant and water treatment agent.
Presidential Candidates for Election
For presidential candidates in the United States, Ireland, Poland, Russia, Trinidad and Tobago, and Uruguay, the minimum age requirement for candidates to be eligible for election is 35 years old.
Social Security Retirement Benefits
Social Security benefits in the United States are calculated based on the 35 highest years of earnings from the candidate's life, which provides financial support to retired individuals who have contributed to the system for self-care.
35mm Film Gauge
35mm film is the most commonly used film gauge for both analog photography and motion pictures due to its durability and wide range of shooting capabilities. It is also the standard gauge for professional cameras and video cameras.
Minimum Age Requirement for Candidates
Candidate eligibility for election to the presidency in countries such as Ireland, Poland, Russia, Trinidad and Tobago, and Uruguay requires a minimum age of 35 years old.–
The number 35 holds a multitude of fascinating properties and has applications in various fields. Whether it's the mathematical properties of tetrahedral numbers, the sum of triangular numbers, or its role in primality testing, 35 is a number with a rich history and diverse uses.
