## SidYuca - In memoriam

## SidYuca - In memoriam

(OP)

I would like to note the passing of Sid Hollander, who briefly posted in this Puzzles for Programmers forum under the handle SidYuca. I recently came across the following obituary:

https://yucalandia.com/2022/05/30/sid-hollander-yu...

In my view Sid's contributions to the Puzzles for Programmers forum were much greater than the short time he participated here. In particular I spent a significant amount of time on two of his threads, learning a lot about smooth numbers and Bertrand's Paradox in the process. See thread1551-1667009: I can Predict your response with '90%' accuracy. and thread1551-1667663: BULLS' EYE- Students A, B & C

https://yucalandia.com/2022/05/30/sid-hollander-yu...

In my view Sid's contributions to the Puzzles for Programmers forum were much greater than the short time he participated here. In particular I spent a significant amount of time on two of his threads, learning a lot about smooth numbers and Bertrand's Paradox in the process. See thread1551-1667009: I can Predict your response with '90%' accuracy. and thread1551-1667663: BULLS' EYE- Students A, B & C

## RE: SidYuca - In memoriam

List all pairs of consecutive positive integers where both numbers in the pair have no prime factor greater than seven.

Solutions are plentiful among small pairs. For example (2,3) and (49,50) are valid solutions, but (50,51) is not. It's less obvious but turns out to be true that there are only a finite number of solutions. I don't currently know the answer, but my online references indicate that the math is within my capability. Naturally I would welcome other participants, but I plan on working on the problem regardless of the level of interest it generates. (Note: I'm fairly sure that all valid pairs can easily be found via a computer search. That's a legitimate way of finding the answer, but it's also possible to find the pairs more systematically and prove that there are no others.)

## RE: SidYuca - In memoriam

## Hidden:

(1,2)

(2,3)

(3,4)

(4,5)

(5,6)

(6,7)

(7,8)

(8,9)

(9,10)

(14,15)

(15,16)

(20,21)

(24.25)

(27,28)

(35,36)

(48,49)

(49,50)

(63,64)

(80,81)

(125,126)

(224,225)

(2400,2401)

(4374,4375)

Perhaps someone will independently verify these results.

## RE: SidYuca - In memoriam

Fortunately Størmer's theorem provides bounds for how large the solutions can be, as well as a method for systematically finding them all. The Wikipedia article is https://en.wikipedia.org/wiki/St%C3%B8rmer%27s_the...

## RE: SidYuca - In memoriam

x

^{2}- 2*y^{2}= 1x

^{2}- 6*y^{2}= 1x

^{2}- 10*y^{2}= 1x

^{2}- 12*y^{2}= 1x

^{2}- 14*y^{2}= 1x

^{2}- 20*y^{2}= 1x

^{2}- 28*y^{2}= 1x

^{2}- 30*y^{2}= 1x

^{2}- 42*y^{2}= 1x

^{2}- 60*y^{2}= 1x

^{2}- 70*y^{2}= 1x

^{2}- 84*y^{2}= 1x

^{2}- 140*y^{2}= 1x

^{2}- 210*y^{2}= 1x

^{2}- 420*y^{2}= 1The first four solutions to each Pell equation generate a possible valid number pair, so we have 4*15 = 60 pairs to test. Not all of these pairs will be valid solutions, but Størmer's theorem guarantees that there are no others. It turns out that the largest pair that needs to be tested is (1040514048, 1040514049)

## RE: SidYuca - In memoriam

largest prime = 7, number of pairs to test = 60

largest prime = 11, number of pairs to test = 186

largest prime = 13, number of pairs to test = 441

largest prime = 17, number of pairs to test = 1143

largest prime = 19, number of pairs to test = 2550

largest prime = 23, number of pairs to test = 6132

## RE: SidYuca - In memoriam