Hey everyone,
I didn’t write a problem yesterday, so today I’m going to write about two famous unsolved problems in mathematics: the Twin Prime Conjecture and Goldbach’s Conjecture. These two problems are so famous because they’re simply stated, yet so seemingly difficult to solve.
Problem 4:
A twin prime is a prime number that differs from another prime number by two. A twin prime can refer to a single prime number with this property, or it can refer to a pair of prime numbers that differ by two. Are there an infinite number of twin primes? The Twin Prime Conjecture states that there are infinitely many primes p such that p+2 is also prime. Is the conjecture true?
Solution 4:
As of today, there are no proofs of the Twin Prime Conjecture accepted by the mathematical society. This is one of the oldest unsolved problems in mathematics and no one (that we know of) has been able to solve it for thousands of years! It’s my wild guess that there are an infinite number of twin primes. Mathematicians have made some progress, and have shown that there are infinitely many n such that at least two of n, n+2, n+4, and n+6 are prime.
Problem 5:
Goldbach’s Conjecture states that every even number greater than two can be written as the sum of two primes. Is Goldbach’s Conjecture true?
Solution 5:
Again, no one knows the answer to this conundrum. There’s a large prize of $1,000,000 (I fathom much more has been spent trying to solve the problem) offered to the solver! The term “solver,” however, is erroneous, because I suspect the final proof will have been the cumulative result of several mathematicians’ works over several centuries.
I don’t want to close this post without solving a problem, so today I’ll write a bonus problem proving the fact that the product of two negative numbers is a positive number.
Problem 6:
Prove that the product of two negative numbers is a positive number.
Solution 6:
If a and b are any numbers, then:
$latex displaystyle (-a) cdot (-b) +[-(a cdot b)] = (-a) cdot (-b) + (-a)cdot b$
(Associative Property)
$latex displaystyle (-a) cdot (-b) +[-(a cdot b)] = (-a) cdot [(-b)+(b)]$
(Distributive Property)
$latex displaystyle (-a) cdot (-b) +[-(a cdot b)] = (-a) cdot 0$
(Since –b + b = 0)
$latex displaystyle (-a) cdot (-b) +[-(a cdot b)] = 0$
(Since –a x 0 = 0)
$latex displaystyle (-a) cdot (-b) = (a cdot b)$
(Add a x b to both sides)
That’s why the product of two negative numbers is a positive number! Ever since kindergarten I’ve taken this fact for granted, but I’m grateful for coming across this proof now, for the sake of better understanding.
Note: I used a new command today, cdot, which specifies the dot commonly used to denote multiplication.