Number theory

Primes, perfect numbers, and the exact statements of Collatz, Goldbach and abc — including the conditions that are usually dropped.

10 questions. Reveal each answer when you have committed to one.

1. Why is 1 excluded from the prime numbers?

   • AIt is not an integer
   • BIt has no divisors
   • CIt is too small to be useful
   • DSo that factorisation into primes is unique
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   D. So that factorisation into primes is unique

   If 1 counted as prime, every number would have infinitely many prime factorisations. The exclusion is a convention chosen to make the fundamental theorem of arithmetic true. More on prime number.

2. Goldbach’s conjecture states that every even integer greater than 2 is

   • Aa sum of two primes
   • Ba product of two primes
   • Ca difference of two primes
   • Da sum of three primes
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   A. a sum of two primes

   This is the binary or strong form and it remains open. The weak form, that every odd number above 5 is a sum of three primes, was proved by Helfgott in 2013. More on goldbach conjecture.

3. In the Collatz rule, an odd number n is replaced by

   • An / 2
   • B3n + 1
   • Cn − 1
   • D3n
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   B. 3n + 1

   Even numbers are halved and odd numbers become 3n + 1. The conjecture is that iterating this always reaches 1. More on collatz conjecture.

4. Over which numbers is the Collatz conjecture stated?

   • AThe primes
   • BAll integers
   • CThe positive integers
   • DThe rationals
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   C. The positive integers

   The restriction is essential rather than tidiness. Over all integers the claim is false: −1 → −2 → −1 cycles forever, −5 enters a five-step cycle, and 0 maps to itself. More on collatz conjecture.

5. A perfect number is equal to

   • Aits own square root
   • Bthe sum of all its divisors
   • Cthe product of its divisors
   • Dthe sum of its proper divisors
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   D. the sum of its proper divisors

   The proper divisors are the positive divisors strictly below the number itself: 6 = 1 + 2 + 3. Whether any odd perfect number exists is open and has been for over two thousand years. More on perfect number.

6. For two numbers to be amicable, they must be

   • Adistinct
   • Bboth even
   • Cboth prime
   • Dconsecutive
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   A. distinct

   Without distinctness every perfect number would pair with itself and the definition would collapse into the previous one. The smallest genuine pair is 220 and 284. More on amicable numbers.

7. The Riemann hypothesis is a claim about which zeros of the zeta function?

   • AThe trivial zeros
   • BThe nontrivial zeros
   • CThe zeros of the gamma function
   • DAll of them
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   B. The nontrivial zeros

   The trivial zeros are the negative even integers and are not at issue. The hypothesis says every nontrivial zero has real part exactly 1/2. More on riemann hypothesis.

8. The trivial zeros of the Riemann zeta function are

   • Athe points with real part 1/2
   • Bthe negative odd integers
   • Cthe negative even integers
   • Dthe prime numbers
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   C. the negative even integers

   They arise from the functional equation, which is also what distinguishes the line with real part 1/2 by reflecting the plane about it. More on riemann zeta function.

9. In the abc conjecture, rad(abc) denotes

   • Athe radius of convergence
   • Bthe largest prime factor of abc
   • Cthe sum a + b + c
   • Dthe product of the distinct primes dividing abc
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   D. the product of the distinct primes dividing abc

   The radical discards exponents entirely, which is the whole content of the conjecture: a sum of numbers built from few distinct primes cannot itself have many repeated ones. More on abc conjecture.

10. The abc conjecture requires the integers a, b and c to be

    • Acoprime
    • Beven
    • Cconsecutive
    • Dprime
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    A. coprime

    Without coprimality the statement is false immediately. It is the condition most often dropped when the conjecture is stated informally. More on abc conjecture.