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OPC 2 Problem 5

\(5. \text{Prove that if } n \ge 4 \quad then \quad n , n+2,n+4 \text{ can't all be primes.}\)


Give different methods of solving this. This was a problem to the OPC 2. The OPC 2 has already ended. OPC 3 will be released soon.Calvin sir suggested me to do this.So Thanks to him.\(\ddot\smile\)

Note by Rajdeep Dhingra
1 year, 9 months ago

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  • Statement1:- A prime number greater than 2 must be odd

  • Statement 2 : - A prime number n when divided by any number d , such that d < n , we get a remainder of either 1 or 2 .

Restating the above statement: .... If

(n =Prime Dividend, q = Quotient, and d = Divisor ) , given 0<d<n

   1.  n = d x q

   2. n=d x  q  + 1

   3. n= dx q +2

Now , for n>= 4 , let d=3
then q must be greater than or at least equals to 1 , ie. q>= 1

case 1: For q > 1

   a) n= 3 x q        {Not Prime}


    b)   n=3x q  +1 ....

      or,    n+2=3 x (q+1)     { Not Prime}

   c) n=3x q  +2 

    or , n+4=3x  (q+2)      { Not Prime}

so we can see that ( n , n+2 and n+4 ) cannot be all primes for each a) , b) and c) since they are product of 3 and a positive integer

case 2 :For q = 1

   a) n=4

    b) n=5

.for n=4 , n+2 and n+4 are not primes , also for =5 , n+4 is not prime .

Hence we can assert that for n>=4 n, n+2, n+4 cannot be all Primes Abhinava Sanyal · 1 year, 9 months ago

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one of them will be divisible by 3. so they cant all be primes Kaustubh Miglani · 1 year ago

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@Kaustubh Miglani Not necessarily.. Nihar Mahajan · 9 months ago

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@Nihar Mahajan Sorry That Was A typo.The

real statement should be one of them will be divisible by 3 so they cant all be primes Kaustubh Miglani · 9 months ago

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@Kaustubh Miglani Hm, thats correct now. Nihar Mahajan · 9 months ago

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Let n be prime, so n is odd.

Case1: n mod 5=1, then n+4 mod 5 =0, so n+4 is not prime. Case2: n mod 5=3, then n+2 mod 5 =0, so n+2 is not prime.

Complete! Archit Boobna · 1 year, 9 months ago

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Comment deleted Mar 29, 2015

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@Rajdeep Dhingra Check my reply Archit Boobna · 1 year, 9 months ago

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Comment deleted Mar 29, 2015

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@Rajdeep Dhingra Why should I tell you here? Archit Boobna · 1 year, 9 months ago

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Comment deleted Mar 29, 2015

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@Rajdeep Dhingra Delete this comment, you are not supposed to tell the answer to my question here Archit Boobna · 1 year, 9 months ago

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@Archit Boobna Fine. Rajdeep Dhingra · 1 year, 9 months ago

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