# Sunday Times Teaser 3003 – All That Glitters

*by Nick MacKinnon*

#### Published Sunday April 12 2020 (link)

My aunt has a collection of sovereigns, and she set me a challenge. “You can have the coins if you can work out the dates, which (in increasing order) are equally spaced and all in the 20th century. The number of coins is an odd prime. The highest common factor of each pair of dates is an odd prime. The sum of the number of factors of each of the dates (including 1 and the date itself) is an odd prime.” I worked out the dates, though the gift was much less valuable than I’d hoped.

What were the dates?

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Using my number theory library (recently updated).

Hi Brian

I would be interested to know how you chose the size of pr_set.

When considering the hcf of pairs of dates it is immediately obvious that the maximum possible value is 97.

When considering the sum of tau values it is not immediately obvious that 2000 is an upper bound.

Best wishes

Tony

Hi Tony, I didn’t do anything sophisticated in choosing 2000 since I didn’t need to. The cost of producing the set is insignificant and the cost of asking if something is in a set in Python doesn’t depend on set size. I knew that 2000 was good for the hcf’s and a small amount of thought convinced me that the sum of the numbers of factors would be well below this maximum (I didn’t do the sum but if all 100 dates were included the sum of the tau values would be 867).