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Is this proof of 4th Landau's problem correct?

Dirichlet's theorem

For any two coprime positive integers \(a\) and \(d\), there are infinitely many primes of the form \(a + nd\), where \(n\) is a non-negative integer.

Argument: There are infinitely many prime of the form \(n^2 + 1\).

Proof:

Let \(n > 0\). For \(n = 1, n^2 + 1 = 2\) which is prime. Otherwise, if \(n^2 + 1 \neq 2\) then \(n^2 + 1 = (n)(n) + 1\) and \(\gcd(n,1) = 1\). Thus, by Dirichlet's theorem there are infinitely many prime of the form \((n)(n) + 1\).

Note by Paul Ryan Longhas
1 year, 2 months ago

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Wait this is still an unsolved problem right? The proof may seem correct because \(n\) is used in two different contexts in the expression \((n)(n)+1\), one \(n\) is given/fixed while the other \(n\) not. Dirichlet can only tell us that there's infinitely many \(m\) such that \(nm+1\) is a prime, but it doesn't guarentee that \(m=n\). Xuming Liang · 1 year, 2 months ago

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Put n=8 Then from n^2+1 We get 8^2 +1 = 64+1 =65 Which is not prime Aaryan Saha · 1 year, 2 months ago

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@Aaryan Saha There are infinitely many primes of the form \(n^2+1\) doesn't mean that all the numbers of the form \(n^2+1\) are primes. For example, there are infinitely many positive integers which can be termed as prime numbers and so there are infinitely many primes. Does this mean that all positive integers are prime? Tapas Mazumdar · 4 months ago

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