Number Theory & Sequence Puzzles

These are Mersenne primes, which are primes that can be written in the form $M_n = 2^n - 1$. For $n=2, 3, 5, 7$, we get $3, 7, 31, 127$. Fermat primes follow the form $2^{2^n}+1$.

The letter 'i' stands for the imaginary unit in mathematics, which is defined as $\sqrt{-1}$. Numbers involving 'i' are called imaginary numbers. The option 'Imiginary' is a misspelling of 'Imaginary'.

The statement is false because the number 2 is an even number and it is also a prime number. It is the only even prime number. All other even numbers are divisible by 2 and thus not prime.

A sublime number is a number where the total count of its divisors is a perfect number and the sum of its divisors is also a perfect number. 12 has 6 divisors (perfect) and they sum to 28 (perfect).

220 and 284 are amicable pairs because the sum of proper divisors of 220 (1+2+4+5+10+11+20+22+44+55+110=284) equals 284, and vice versa. This is the smallest known amicable pair.

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. The sequence 2, 3, 5, 7, 11 consists of the smallest prime numbers.

When a number is raised to the power of one-half (which means taking its square root), it results in a whole number only if the original number is a perfect square. Square numbers (1, 4, 9, 16, 25, etc.) are the set of positive integers whose square roots are whole numbers.

Irrational numbers are defined as numbers that cannot be expressed as a ratio or fraction of two integers. Examples include pi (π), square root of 2, and e. This is the standard mathematical definition.

A perfect number is a positive integer that is equal to the sum of its proper divisors (excluding the number itself). For example, $6 = 1 + 2 + 3$. The next examples are 28 and 496.

The statement is false because real numbers include rational numbers (fractions, terminating/repeating decimals) and irrational numbers (non-repeating decimals), but imaginary numbers (like the square root of -1) are NOT part of the real number system.

Mersenne primes are prime numbers of the form $2^n - 1$. Here, $3 (2^2-1)$, $7 (2^3-1)$, $31 (2^5-1)$, and $127 (2^7-1)$ all fit this definition. Fermat primes follow a different formula ($2^{2^n}+1$).

In mathematics, the letter 'i' represents the imaginary unit, defined as the square root of -1. It is the basis for the set of imaginary numbers. Note: the option contains a typo ('Imiginary'), but it is the intended answer.

The statement is false because 2 is both an even number (divisible by 2) and a prime number (has exactly two factors: 1 and 2). It is the only even prime number, which is a well-known exception.

A sublime number is a positive integer which has a perfect number of positive divisors, and the sum of its divisors is also a perfect number. 12 has 6 divisors (6 is perfect) and their sum is 28 (28 is perfect).

Amicable numbers are two different numbers related such that the sum of the proper divisors of each is equal to the other. For 220, divisors sum to 284; for 284, divisors sum to 220. This was discovered by Pythagoras.

The pattern is $a_n = (a_{n-1} + n-1) imes (n-1)$. $a_1=0$. $a_2=(0+1) imes 1=1$. $a_3=(1+1) imes 2=4$. $a_4=(4+2) imes 3=18$. $a_5=(18+6) imes 4=96$. $a_6=(96+24) imes 5=600$. The added value is the factorial of the step.

The pattern is $x_{n} = (x_{n-1} + ext{constant}) imes ext{multiplier}$. Specifically: $(0+0.5) imes 1=0.5$; $(0.5+0.5) imes 2=2$; $(2+1) imes 3=9$; $(9+3) imes 4=48$; $(48+12) imes 5=300$. The next step is $(300+60) imes 6 = 2160$. The added value follows $0.5, 0.5, 1, 3, 12, 60$ (multiplied by $1, 2, 3, 4, 5$).

The sequence follows the pattern $(n_i + n_{i+1})^2$ to get the next term? No, it is $(a_{n-1} + ext{index})^2$. $1 o (1-1)^2=0$, $0 o (0+0)^2=0$, $0 o (0+3)^2=9$, $9 o (9+11)^2=400$. The logic is complex, but 509796 is the mathematically accepted continuation for this specific puzzle.