Special whole numbers...

[Jens K Andersen]

Originally Posted by Richard Cameron

232 is the sum of two squares (6^2 + 14^2); 233 is the sum of the squares of 8 and 13; and 234 is the sum of the squares of 3 and 15. 232-233-234 is the smallest such trio, which does seem interesting to me: should be in the wikipedia entry.

16 = 0^2 + 4^2, 17 = 1^2 + 4^2, 18 = 3^2 + 3^2
72 = 6^2 + 6^2, 73 = 3^3 + 8^2, 74 = 5^2 + 7^2
80 = 4^2 + 8^2, 81 = 0^2 + 9^2, 82 = 1^2 + 9^2
144 = 0^2 + 12^2, 145 = 1^2 + 12^2, 146 = 5^2 + 11^2
232 = 6^2 + 14^2, 233 = 8^2 + 13^2, 234 = 3^2 + 15^2

16, 17, 18 is first if all squares are allowed.
72, 73, 74 is first if 0^2 is disallowed.
80, 81, 82 is first if a repeated square is disallowed.
232, 233, 234 is first if both 0^2 and a repeated square are disallowed.
Too many conditions to be that special to me.

[cheesehead]

Originally Posted by Jens K Andersen

80 = 4^2 + 8^2, 81 = 0^2 + 9^2, 82 = 1^2 + 9^2

< snip >

232 = 6^2 + 14^2, 233 = 8^2 + 13^2, 234 = 3^2 + 15^2

< snip >

80, 81, 82 is first if a repeated square is disallowed.

... depending on the definition of repeated, that is: 9^2 occurs in both 81 and 82.

232, 233, 234 is first if both 0^2 and a repeated square are disallowed.

No repeats of any kind: 3^2, 6^2, 8^2, 13^2, 14^2, 15^2

[petrw1]

How about 337 ...

[Richard Cameron]

sorry for the delay in replying

My answer to the original posting of 232 (by Mally Goode, back last August) was certainly what he had in mind as his previous answer (to the number 239) was taken from the same page of the same book.

David Wells in that book (The Penguin Dictionary of Curious and Interesting Numbers) doesn't give a source for 232. He does say that the sum is the 'hypotenuse of a pythagorean triangle' which i read as excluding 0 as a case, but allowing repeats. So 72-73-74 fits.

Not one of Wells's clearer entries. Looks like i'll have to write to Penguin (again!)


Richard

[mfgoode]

Originally Posted by petrw1

How about 337 ...

337 -< permutable prime


A permutable prime is a prime number, which, in a given base, can have its digits switched to any possible permutation and still spell a prime number. H. E. Richert, who supposedly first studied these primes, called them permutable primes[1], but later they were also called absolute primes[2].

In base 10, the all permutable primes with less than 4 digits are (with the permutations listed in parentheses):

2, 3, 5, 7, 11, 13(31
There are a few more less than 4 digits

233

Mally

[petrw1]

Originally Posted by mfgoode

337 -< permutable prime
In base 10, the all permutable primes with less than 4 digits are (with the permutations listed in parentheses):
Mally

Is this to suggest there are NONE with 4 or more digits?

[mfgoode]

Of course there are more but I just forgot the values hence gave the gentle hint. If you want more details on such numbers Google 'permutable primes'. It provides some very interesting facts on such numbers.

Mally

[Richard Cameron]

Originally Posted by petrw1

Is this to suggest there are NONE with 4 or more digits?

You need look no further than wikipedia to read
"There is no n-digit permutable prime for 3<n<6*10^175" [excluding the repunits of course] . I could get into trouble for saying this, but to me that DOES suggest they don't exist above 3 digits.

Richard

[mfgoode]

Good observation and conclusion.

Permutable primes are primes with at least two distinct digits which remain prime when permuting the digits. Permutable primes are also circular primes.

The only permutable primes up to 466 digits are

13, 17, 37, 79, 113, 199, 337

and their permutations. It is very unlikely that there are other permutable primes.

Mally

[10metreh]

Why has this thread been inactive for so long? I think it is great.

233 is the smallest prime factor of 2^29-1.

4897256

[davieddy]

Originally Posted by 10metreh

Why has this thread been inactive for so long? I think it is great.

See post #363.
Some threads die a natural death and it was perhaps
only Mally (mfgoode) who kept flogging it.
He died in September 2007 as you may know.
David