[ October 9, 2001 ]
An ODD_DDO puzzle question
1 3 5 7 9

This WONplate is totally dedicated to integers
composed of only ODD digits and whose prime factors are iDem DitO !
Is this a worthy challenge for you ?

Sequence A062016 will give you a good starting point
for the semiprime case (with exactly two prime factors).
Bear in mind that more than two prime factors are allowed.

Note that I am looking for ever larger integers/prime factors
that are sufficiently random. I don't want to highlight the obvious ones
that could be found too easily by reverse engineering like for instance
3 * Repunit(n).
or that could be derived from an infinite pattern like
(9)_n5 = 5 * 1(9)_n
Lastly if the digitlength of the factors are > 1 and near equal to each
other it certainly will enhance the beauty of the all-odd number found !

Solutions

Two prime factors
Integer ?_{all_odd} = f1 * f2
(possible if f1 and f2 > 7 ?)

Three prime factors
7335737 = 191 * 193 * 199
15995731573 = 1153 * 1933 * 7177

Four prime factors
111333 = 3 * 17 * 37 * 59

Five prime factors
7977553773 = 3 * 7 * 17 * 113 * 197753

Etc.

[ June 5, 2002 ]
Jean Claude Rosa (email) solved the small question for 2 factors.
The answer is no. Every prime factor larger than 7 and 'all odd'
always ends in one of the following 20 numbers :
11, 13, 17, 19, 31, 33, 37, 39, 51, 53,
57, 59, 71, 73, 77, 79, 91, 93, 97 or 99.
If we multiply two by two any pair of these 20 numbers
we get the result that
The digit at the tens position is always even.
So for the first case with two prime factors there exist no solution
for factors larger than 7 of course.
Thus the products of two prime factors 'all odd' ends in
01, 03, 07, 09, 21, 23, 27, 29, 41, 43,
47, 49, 61, 63, 67, 69, 81, 83, 87 or 89.
If we next multiply two by two any pair of these 20 numbers
we still get the same result
The digit at the tens position is always even.
This means there is also no solution for the cases
with 4, 6, 8, ... prime factors 'all odd' and larger than 7.

Additional Question from Jean Claude Rosa [ June 10, 2002 ]
" Is it possible to find a palindromic number 'all odd' as a result
of the product of 3, 5, ... prime factors 'all odd' ? "

[ July 15, 2002 ]
Jean Claude Rosa calculated all the products of 3, 5 and 7
prime factors 'all odd' taken amongst 10 consecutive 'all odd'
prime numbers from 3 up to 3,999,999,979.
Here are the largest 'all odd' composites that he found

With 3 prime factors
391133599733953317333733 =
73131517 * 73131599 * 73133351

With 5 prime factors
775793373119 =
191 * 193 * 197 * 317 * 337

With 7 prime factors
3353535515 =
5 * 11 * 17 * 31 * 37 * 53 * 59

He also found some cute little 'all odd' palindromes !

595 = 5 * 7 * 17
3553 = 11 * 17 * 19
19591 = 11 * 13 * 137

" J'espère avoir un peu de temps pour continuer
à rechercher des palindromes plus grands... "

[ July 16, 2002 ]
This afternoon I (J. C. Rosa) discovered larger 'all odd'
palindromes, which of course made me very happy...
Here they are

With 3 prime factors
9979531359799 = 19 * 71171 * 7379951

With 4 prime factors
9935335335399 = 3 * 191 * 5171 * 3353153

With 5 prime factors
9997159517999 = 19 * 53 * 59 * 937 * 179579

With 6 prime factors
9919959599199 = 3 * 13 * 17 * 59 * 317 * 799991

He also found various palindromic composites with
prime factors NOT all different

19399999391 = 59 * 59 * 5573111
19975357991 = 53 * 53 * 7111199

And finally this curious one !

777555777 = 3 * 37 * 73 * 95959

[ July 21, 2002 ]
J. C. Rosa communicates that the two largest
15-digit 'all odd' palindromes consisting
of 3 or 5 distinct prime factors are

993957131759399 = 977 * 13537 * 75153751

995393999393599 = 13 * 19 * 53 * 193 * 393971573

[ October 21, 2002 ]
Edwin Clark (email) (site) looked for 'all odd' palindromes
with 'all odd' & 'palindromic' prime factors using Maple.

9 = 3 * 3
33 = 3 * 11
393 = 3 * 131
939 = 3 * 313
33933 = 3 * 11311
39993 = 3 * 13331
55 = 5 * 11
77 = 7 * 11
99 = 3 * 3 * 11
1331 = 11 * 11 * 11
3993 = 3 * 11 * 11 * 11
3773 = 7 * 7 * 7 * 11
17377371 = 3 * 3 * 11 * 191 * 919
5775 = 3 * 5 * 5 * 7 * 11

Clearly some of these generalize. For example when 1333...3331 is
a prime we get one of the form 3999...9993. The largest prime of
this form I found was one with 95 digits. [ For this I used Maple's
'isprime' function to determine primality.] Unfortunately all of these
have at least one small prime factor.

I found these by putting in a list the 43 odd palindromic primes
less than 10^5 and found all products of 2, 3, 4 and 5 of these. Then
I checked the products for oddness and palindromicity.

p.s. primes of the form 1333...3331 are PDP primes
Plateau and Depression Primes
and are since then more extensively researched.
The second largest known probable prime of the form
(12*10^n-21)/9 or 4*(10^n-1)/3-1 is 1(3)₁₇₉₅₅1
thus creating an all odd palindrome of 17957 digits !