Problems & Puzzles: Puzzles

Puzzle 908. Descending residues

Vic Bold sent the following nice puzzle.

For two consecutive primes, (p, q) such that q>p. q=nextprime(p) we construct the following scheme of descending residues:

Let pq = p*q

pq mod (p+q)=r1
pq mod r1=r2
pq mod r2=r3
...
pq mod rk-1= rk =1 (stop)

or in a compact form:

(p, q)-> {r1, r2, ...rk}, rk=1

Example by Vic: (p, q) = (11, 13), steps = 5

11*13 mod (11+13) = 23
11*13 mod 23 = 5
11*13 mod 5 = 3
11*13 mod 3 = 2
11*13 mod 2 = 1 (stop)

or (11, 13) -> {23, 5, 3, 2, 1}

In this example all the five residues are primes or 1.

Carlos Rivera found a second example like the above such that steps = 6 and all the six residues are primes or 1: (53, 59) -> {103, 37, 19, 11, 3, 1}

Rivera suggests that there are no larger scheme of pure primes or 1 residues, if p & q are consecutive primes.

Q1. Can you find another pair of consecutive primes (p, q) with more that six residues prime or 1 and none composite?
 

Q2. Can you find another pair of primes (p, q) not necessarily consecutive, with more than six residues prime or 1 and none composite?

Rivera has gotten a pair of consecutive primes (p, q) with 20 primes or 1, in a sequence of 40 descending residues:

(3810860329, 3810860351) -> { 7621720559, 1905433679, 49252203, 30653789, 30177730, 20955199, 15150616, 10199407, 3078304, 2126007, 1455269, 1295988, 1132283, 674812, 553703, 449111, 278067, 156221, 79518, 42239, 12303, 6737, 2309, 1777, 1631, 719, 607, 281, 189, 131, 59, 48, 23, 17, 6, 5, 4, 3, 2, 1}; primes & 1 in the set are in bold type.

Q3. Send another example of consecutive primes (p, q) with more than 20 residues prime or 1 and some composites.


 


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