Problems & Puzzles: Puzzles

Puzzle 145.  The ISPP & SSPP functions

Hans Gunter from Koln (Germany), sent the following puzzle:

Let n >= 2. As a generalization of the integer part of a number one defines the Inferior Smarandache Prime Part as: ISPP(n) is the largest prime less than or equal to n.

For example: ISPP(9)=7 because 7<9<11, also ISPP(13)=13.

Similarly the Superior Smarandache Prime Part is defined as: SSPP(n) is smallest prime greater than or equal to n.

For example: SSPP(9)=11 because 7<9<11, also SSPP(13)=13.

Questions:

Let k > 0 be a given integer. Solve the diophantine equation:
ISPP(x) + SSPP(x) = k.

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References:

C. Dumitrescu & V. Seleacu, "Some Notions And Questions In Nimber Theory", Sequences 37 \$ 38, http://www.gallup.unm.edu/~smarandache/SNAQINT.txt

Tatiana Tabirca & Sabin Tabirca, "A New Equation For The Load Balance Scheduling Based on Smarandache f-Inferior Part Function", http://www.gallup.unm.edu/~smarandache/tabirca-sm-inf-part.pdf

[The Smarandache f-Inferior Part Function is a greater generalization of ISPP.]

Solution

Partial solutions came from Jean Marie Charrier (Lyon, France), Teresinha DaCosta (Bello Horizonte, Brasil) and Rene Blanch (France). Only Richard Kelley sent (in two tries) the complete solution. Paraphrasing the Kelly's lines:

a) if k is the sum of two consecutive primes p1 & p2 then x takes all the values of the open interval (p1, p2)
b) if k=2.p then x=p
c) otherwise there is not any solution

Notice that if p1 & p2 are 2 & 3 then applies the rule a), that is to say there is not any x value solution of the discussed equation because there is not any value in open interval (2, 3).

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Jim Howell also found a complete and independent solution for this puzzle.

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