Number Theory

All primes greater than 3 can be expressed as 6n±1 because numbers of form 6n, 6n±2, 6n±3 are divisible by 2 or 3. For example: 7=6(1)+1, 11=6(2)-1, 13=6(2)+1. The pattern fails for 5n±1 and 7n±1.

Phyllotaxis, the arrangement of leaves on a stem, often follows a pattern related to Fibonacci numbers. This mathematical sequence (1, 1, 2, 3, 5, 8...) describes the spiral patterns found in many plants, which optimizes space and exposure to sunlight.

A prime number is divisible only by 1 and itself. 18 is divisible by 1, 2, 3, 6, 9, and 18, so it has multiple divisors and is not prime.

19 is a prime number because its only divisors are 1 and 19. It cannot be divided evenly by any other integer, making it a true prime.

Let the four primes in ascending order be p1, p2, p3, p4. Since p1×p2×p3 = 385 and p2×p3×p4 = 1001, we factor 385 = 5×7×11 and 1001 = 7×11×13. The middle two primes 7 and 11 overlap, so p2 = 7, p3 = 11, and thus p4 = 13, which is the largest prime.

If x and y are prime, their difference can be 1 (3-2) or 3 (5-2). It can be 15 (17-2) or 9 (11-2). However, for a difference to be 23, one number must be even (2) and the other 25, which is not prime. If both are odd, the difference must be even.

2 is the smallest prime number. A prime must be greater than 1 and only divisible by 1 and itself. Neither 0 nor 1 qualifies as prime - 0 has infinite divisors, and 1 only has one divisor.

A prime number is only divisible by 1 and itself. 641 is prime (not divisible by 2, 3, 5, 7, 11, 13, 17, 19, or 23). 791 = 7×113, 687 = 3×229, and 323 = 17×19 are all composite.

A perfect number is a positive integer equal to the sum of its proper divisors (excluding itself). The smallest perfect number is 6 (1+2+3=6). Option A describes an abundant number. Option B describes prime numbers. Option D describes irrational numbers. Only option C correctly defines perfect numbers, a concept dating back to Greek mathematics.

Irrational numbers are real numbers that cannot be expressed as a simple fraction or ratio of two integers (like $\pi$ or $\sqrt{2}$). Rational numbers, by contrast, can be expressed as ratios.

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 $2^{2^n}+1$.