Problems & Puzzles: Puzzles

Puzzle 942. Semiprimes and Pascal Triangle

Vic Bold sent the following nice puzzle:

Given a semiprime N, construct the first four rows of a Pascal Triangle using its four divisors as shown in the example. Compute the sum S of all the elements of these four rows. Is S prime?

Example: N=14. Divisors: 1, 2, 7, 14

 12  2 7  4  7 14  11  11  14

S(14)=73, Prime!

Q. Are consecutive semiprimes N & N+1 such that S(N) and S(N+1) are both prime numbers? If so, send your largest example. If not, explain why.

Contributions came from Fausto Morales, Jan van Delden, Emmanuel Vantieghem

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Fausto wrote on Feb 9, 2019:

No examples are possible.

Let p and q denote prime numbers, with p < q.

Note first that a semiprime s with S(s) prime is not of the form 3p since then
S(s) = 10p + 25 = 5(2p + 5), not prime.

Then s = 1 (mod 3) and s+1 = 2 (mod 3).

 Case 1: s even.
s = 2p -> p = 2 (mod 3) and S(s) = 8(p+2) + 1 is divisible by 3 (not prime).

 Case 2: s odd.

Then s = 1 (mod 6), (not 3 by the initial observation, and not 5, by the initial observation applied to s+1).

It follows that  p and q are (I) both congruent to 1 (mod 6) or (II) both congruent to 5 ( mod 6).

Note that S(s) = (2p+4)q + 8p + 1.

(I) and (II) can be ruled out, as they both make S(s) divisible by 3.

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Jan wrote on Beb, 9, 2019:

These semiprimes N,N+1 dont exist.

I assume that p<q, with N=pq.

The divisors are in {1,p,q,pq} and S(p,q)=1+8p+4q+2pq.

We have:

S(p,q) mod 3 = 1  p + q  pq mod 3

If both N and N+1 are semiprimes we must have that one of these two semiprimes is divisible by 2.

S(2,q) mod 3 = -1  q, q=1 mod 3pq=2q=-1 mod 3
[If q=3, we have either N or N+1 equal to 6, but 5 and 7 are not semiprimes].

S(3,q) mod 3 = 1  2q mod 3, q=1 mod 3, pq=3q=0 mod 3.

S(1 mod 3,q) mod 3= q  q = 0 mod 3, hence S(p,q) is not prime

S(-1 mod 3,q) mod 3 = 2 + 2q, q=1 mod 3, pq=-1 mod 3

If 2|N+1 we have N+1=-1 mod 3 and N=1 mod 3 which is impossible
If 2|N we have N=-1 mod 3 and N+1=0 mod 3

If a solution exists we must have:

2q[1]+1=3q[2] with q[1],q[2] both 1 mod 3, 1 mod 2 thus 1 mod 6

If we choose q[1]=6k+1,q[2]=6l+1 and solve 2q[1]+1=3q[2], or

2(6k+1)+1=3(6l+1), 12k=18l, or 2k=3l, giving:

q[1]=9l+1, q[2]=6l+1, l even, giving q[1]=18m+1, q[2]=12m+1

S(2,q[1])=144m+25, S(3,q[2])=120l+5 which is always divisible by 5
S(3,q[2]) is not prime, no solution

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Emmanuel wrote on 11 Feb, 2019:

(It is clear that Vic does not consider squares of primes as semiprimes since he speaks of four divisors)

We have, when  p  and  q  are primes with  p < q : S(pq) = 1 + 8p + 4q + 2pq.

Suppose  N  is a semiprime > 6  and divisible by  3.
Thus, N = 3q, where  q  is a prime number > 3.
But then  S(N) = 1 + 24 + 4q + 6q == 0 mod 5.
Thus, when two semiprimes are consecutive numbers then none of these numbers can be divisible by 3.
Let  pq  be an odd semiprime with  3 < p < q  for which  S(pq)  is prime.

Then we have : S(pq) == 1 + 2p + q + 2pq == (1 + 2p)(q + 1) !== 0 (mod 3).
Thus, 1 + 2p  and  q + 1  are not divisible by  3.  But that means : p == 2 (mod 3)  and  q == 1 (mod 3).
Thus, pq == 2 mod 3.
So, the successor of  pq  (i.e. ; pq+1) cannot be a semiprime, since it is divisible by 6.
If the predecessor of  pq  (i.e. : pq-1) is semiprime, it must be of the form  2r, with  r  a prime > 3  and == 2 mod 3.
But then : S(2r) = 1 +16 + 4r + 4r \[Congruent] 2 + 1 === 0 mod 3.  Conclusion : S(2r)  is not prime.
So, the S-value of two semiprimes that are consecutive cannot be twice prime.

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