Landau’s problems on primes

On the Number ofp'-Degree Characters in a Finite Group

“…it is an open problem first posed by Landau whether there are infinitely many primes p with √ p − 1 an integer (see e.g. [13,Sec. 19]).…”

Reasoning About Primes (II)

“…The term twin prime was coined by Paul Stäckel in the late nineteenth century. On the importance of the twin prime conjecture listen to Pintz [2009] In his invited address at the 1912 International Congress of Mathematicians, held in Cambridge, Edmund Landau (1912) gave a survey about developments in the theory of prime numbers and the Riemann zeta-function. Besides this he mentioned (without any further discussion) four specific problems about primes which he considered as "unattackable at the present state of science".…”

“…8 he mentioned them together with the Riemann Hypothesis, using the following words: "After a comprehensive discussion of Riemann's prime number formula we might be some day in the position to give a rigorous answer on Goldbach's Problem, whether every even number can be expressed as the sum of two primes, further on the problem whether there exist infinitely many primes with difference 2 or on the more general problem whether the diophantine equation (2.17) ax+by+c=0 is always solvable in primes x,y if the coefficients a,b,c are given pairwise relatively prime integers." 2 Pintz [2009] discussion of ties between Landau's problems can be summarized like this: For each even m there such that p − p = m such that p + p = m exists at least one pair of primes Maillet (1905) Goldbach (1843) exists at least one pair of consecutive primes 'consecutive existence conjecture' (term by us) (possible for special cases only)…”

Groups with few conjugacy classes

“…It is an open problem of Landau whether there are infinitely many primes p with the property that p − 1 is a square. For more information see Section 19 of [11].…”