Problems & Puzzles: Puzzles

Puzzle 1188  Sum of product of primes step 2 Sebastián Martín Ruiz sent the following puzzle: Let be the sum of the products of primes, step 2. For example: For the first 5 primes, that is, 2,3,5,7,11, we multiply in descending order, step two, and add: 11*5*2+7*3 =110+21=131 prime. 1) 3 + 2= 5 Prime 2) 5·2 + 3= 13 Prime 3) 7·3 + 5·2= 31 Prime 4) 11·5·2 + 7·3= 131 Prime 5) 13·7·3 + 11·5·2= 383 Prime 6) 17·11·5·2 + 13·7·3= 2143 Prime 7) 19·13·7·3 + 17·11·5·2= 7057 Prime 8) 23·17·11·5·2 + 19·13·7·3= 48197 Prime 9) 29·19·13·7·3 + 23·17·11·5·2= 193433 Prime 10) 31·23·17·11·5·2 + 29·19·13·7·3= 1483733 Prime 11) 37·29·19·13·7·3 +31·23·17·11·5·2= 6898961 Composite Q) Make combinations of products and sums of primes that obtain a series with a greater number of successive primes.

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From Set 7-13, 2024, contributions came from Oscar Volpatti

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Oscar wrote:

I attempted some variations of Sebastian Martin Ruiz's process.

 

The first n primes, considered in descending order, are partitioned into two disjoint sets X and Y, according to some given pattern. 

Let integer x be the product of primes in X and let integer y be the product of primes in Y.

The candidate prime is given by the weighted sum z=a*x+b*y. 

 

Choices by Sebastian Martin Ruiz.

Step-2 pattern: XYXYXY...

Unit weights: a=1, b=1, z = x+y.

10 primes found, from index n=2 to index n=11.

Sets X and Y are both non-empty only starting from n=2, but we can use the empty product convention (EPC) for n<2:

the result of multiplying no factors is by definition equal to 1.

Under EPC convention, the given process generates 12 primes, starting from index n=0:

n=0, x=1, y=1, z=2, prime;

n=1, x=2, y=1, z=3, prime. 

 

Variation I.

Step-2 pattern.

Non-unit weights: z=14753*x+257*y.

15 primes found, from n=2 to n=16.

No benefit from EPC, composite values for n=0 and for n=1.

44773, 148301, 312383, 1628227, 4055839, 27658271, 77004401, 635859589, 2230244089, 19708981141, 82452709873, 807913591937, 3544781203199, 37966217331811, 187789109244727.

 

Variation II

Unit weights.

Pattern: XYYYXXXYXYYXYXXXYY (step-18?).

17 primes found, from n=2 to n=18.

19 primes found under EPC, from n=0 to n=18.

2, 3, 5, 11, 37, 127, 463, 1511, 6857, 30307, 161377, 909071, 5586787, 40320767, 255517099, 1580041901, 14001904771, 109181551597, 910548512609.

 

Variation III.

Unit weights.

Pattern: XXYXYXYYYXYXYXXXYYXY (step-20?).

18 primes found, from n=3 to n=20 (n=3 least index with sets X and Y both non-empty).

21 primes found under EPC, from n=0 to n=20.

2, 3, 7, 17, 73, 241, 1451, 4751, 18311, 45377, 275393, 967229, 6139423, 34886237, 232888157, 1703343263, 19596411823, 143929975841, 1156346371769, 11039423608261, 76726934417459.

 

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