Understand the problem

Let \(p_1,p_2,p_3\) be primes with \(p_2\neq p_3\), such that \(4+p_1p_2\) and \(4+p_1p_3\) are perfect squares. Find all possible values of \(p_1,p_2,p_3\).

 

Start with hints

Do you really need a hint? Try it first!

Let \(4+p_1p_2=m^2\) and \(4+p_1p_3=n^2\), where \( m,n \in \mathbb{N}\).

\(\Rightarrow p_1p_2=(m-2)(m+2)\) and \(p_1p_3=(n+2)(n-2)\).

Since \(p_1,p_2,p_3\) are primes with \(p_2\neq p_3\) \(\Rightarrow m\neq n\).

Case –I: \(p_2<p_3 \Rightarrow m<n\).

Clearly, \(p_1=m+2=n-2, \Rightarrow n=m+4\)

\(p_2=m-2\) and

\(p_3=n+2=m+6\).

Therefore, \((m+2),(m-2),(m+6)\) are all prime numbers.

We see that \(m=5\) satisfy the above condition. And then,\( p_1=7,p_2=3,p_3=11\) .

 

 

Case-II: \(p_2>p_3 \Rightarrow m>n\).

\(\Rightarrow p_1=m-2=n+2 \Rightarrow m=n+4\)

\( p_2=m+2=n+6\) and

\(p_3=n-2\).

Now \((n+2),(n+6), (n-2)\) all are primes. Again , \(n=5\) satisfy this condition. Hence \(p_1=7,p_2=11,p_3=3\).

 

Thus all possible values of \(p_1,p_2,p_3\) are ( 7,3,11) and (7,11,3).

Now need to conclude that there does not exist any more triple of prime numbers satisfying the given condition.

Consider these numbers:

\(p_1=m+2,p_2=m-2,p_3=m+6\) , now the gaps between \(p_1,p_2,p_3\) are given by:

\(p_1-p_2=4, p_3-p_2=8 \) and \(p_3-p_1=4\).

We see that for \(m>9\) these gaps are greater than 8 . And between 1 to 9 only \(m=5\) satisfy the given condition. Hence there does not exist any more triples.

 

Connected Program at Cheenta

Source of the problem

I.S.I. (Indian Statistical Institute) B.Stat/B.Math Entrance Examination 2017. Subjective Problem no. 6.

Topic

Number Theory

Difficulty Level

8.5 out of 10

Suggested Book

Elementary Number Theory by David M. Burton

I.S.I. & C.M.I. Entrance Program

Indian Statistical Institute and Chennai Mathematical Institute offer challenging bachelor’s program for gifted students. These courses are B.Stat and B.Math program in I.S.I., B.Sc. Math in C.M.I.

The entrances to these programs are far more challenging than usual engineering entrances. Cheenta offers an intense, problem-driven program for these two entrances.

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