Problems & Puzzles: Puzzles

Puzzle 1056 Follow-up to Puzzle 1055 The current record solution to Puzzle 1055 was obtained by Emmanuel Vantieghem with the prime number 553735009 meaning "gooseless". This s a valid solution because "gooseles" has a definition well established somewhere (in the English wiktionary) as "Without geese" Q. Can you find a larger and valid prime number solution to Puzzle 1055?

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Contributions came from Emmanuel Vantieghem, Giorgos Kalogeropoulos, Carlos Rivera, Ivan Ianakiev, Oscar Volpatti

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Emmanuel wrote:

About my contribution to puzzle1055 : the words that I found came from Tanya Khovanova's Word Game.
 

There, I asked for all the words (with min 2 and max 10 letters) that can be found using the letters from  "hhhiiieeelllgggooosss", using the biggest dictionary (of  295815 words).
 

So, I must adjust the assertion that "most words are not English" (it is still true that  liege  is a town in Belgium, but it has a meaning in English and that  lille  is a town in France).

For puzzle 1056, I revisited Tanya's website and entered the word  "hhhheeeeiiiillllggggoooossss" and asked for "subwords" of min 10 and max 15 letters.
There was only one word : "shigellosis".  Turned upside down, it reads  51507739145, not prime.

So, I'm convinced that there is no bigger English word than  "gooseless" (553735009) ...

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Giorgos wrote:

Some other words that lead to bigger primes than "gooseless" are:

"GLOBELESS" -> 553738079 (prime)

"GOBBLELESS" -> 5537378809 (prime)

"GIGGLELESS" -> 5537379919 (prime)

A definition for "globeless" can be found here: https://en.wiktionary.org/wiki/globeless

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Carlos Rivera wrote:

First, I must advance a warning. In order to deal properly with this puzzle I think that we need to make a distinction between "UPPER CASE" and "lower case" letters. Here is my proposal:

Integer	Upper	Lower	Comment
0	O	o	Both
1	I	l (Lower case from L, not an i)	Distinct
2	None	None	None
3	E	None	Unique
4	None	h	Unique
5	S	s	Both
6	None	g	Unique
7	L	None	Unique
8	B	None	Unique
9	G	b	Distinct
Totals	7	6

Accordingly, when we are looking by words and its associated primes, we must specify if we are using upper or lower caps letters.

You are not able to mix both type of letters except when the word is a proper name. In such a case the first letter could be an upper case letter while the rest of the letters are small caps.

Please observe that using only small caps you have only one vowel ("o") and a total of six distinct letter. This means that in principle the valid words using this type of letter must be shorter than using upper caps, where you count with three vowels ("O", "I" and "E") and a total of seven distinct letters.

After said that, the results I'm giving ahead are using only upper caps letter:

The largest word in Spanish giving a prime, reported by Ivan Ianakiev in Puzzle 1055 was "GOLOSOS" (5050709, 7 digits).

Using a web anagram tool and looking for anagrams of the string "LLLEEEGGGOOOSSSIIIBB", I found two larger primes, 9 digits each:

a) For "ELIGIESES" -> 535319173 (conjugación de elegir, 2ª persona singular del pretérito imperfecto de subjuntivo de elegir, verbo, e-li-gie-ses)

b) For "ELIGIBLES" -> 537819173 (plural de eligible, adjetivo, e-li-gi-bles)

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Ivan wrote:

I doubt whether valid words longer than GOOSELESS exist in English, but using word combinations we can form spaceless and punctuationmarkless expressions/sentences, such as GOGOL/LOSES/GOOGOL/GOOSE/ EGGS or LEGLESS/LEO/SELLS/EEL/ EGGS, which unfortunately does not convert to a prime.

One valid, although not very meaningful, sentence (command) is GO/LEGLESS/ LESSEES, where the respective prime is 5,335,537,553,793,709.

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Oscar wrote:

About puzzle 1056, I suggest the 11-digit prime 37181693771.
I relied on Wikipedia page about calculator spelling:
 

https://en.wikipedia.org/wiki/Calculator_spelling
also referenced within explanation of puzzle 1055:
https://elgeek.com/1021/379009-el-numero-primo-de-google
Each digit may be mapped to one letter, so that number 1234567890 spells "OGBLgShEZI" when turned upside down on a calculator.
 

Then prime 37181693771 spells "ILLEGgIBILE", an Italian word meaning "illegible", "unreadable".

It would have been nicer to use only one capitalization choice between "g" and "G", but both resulting numbers are composite.
 

Interestingly, for the English translation "illegible", both variations 378163771 and 378193771 are prime too (but not long enough to beat Emmanuel Vantieghem's record solution).

 

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