This
palindromic prime number reads the same upside down or when viewed in a
mirror.
Contents
Primes from
Factorials !

Factorial n (n!) means 1 x 2 x 3 x … x
n Unfortunately the next factorial in this series results
in a composite number.
The above shows the number 1 as a prime, although it is normally
considered neither prime nor composite.

PRIME is Prime
Assign the value 1 to A, 2 to B, 3 to C, . . . , 26 to Z. Then

i.e. 16 + 18 + 9 + 13 + 5 = 61 
Sum of 5 and 7 Primes
= a prime
More Prime series
Above is shown three of the five series that use 2, the only even prime
number.
Then I show one of each odd series from three to twentyone. There are a
total of sixtyone series with an odd number of primes (using primes < 100).
every prime < 89 is the leading term in at least one series.
Charles w. Trigg, JRM
18(4),198586, p.247248
Near Repdigit Primes

All primes!
The next prime number in this series, though, is 17 threes with a
one at the end.
Some other numbers in this series (with less then 1800 threes) are:
3_{39}1 (that's 39 threes with a one at the end
3_{783}1
3_{1731}1
Near Repdigit Primes consist of a series of the same digit, then one
different digit; or one digit and then a series of a different
digit.
The above series is particularly attractive because the number of
threes in the first seven primes increase by one. There are only
eleven other primes in this series with less then 1800 threes. 

All primes!
The next prime number in this series, though, is 17 threes with a one at
the end.
Some other numbers in this series (with less then 1800 threes) are:
3_{39}1 (that's 39 threes with a one at the end
3_{783}1
3_{1731}1
Near Repdigit Primes consist of a series of the same digit, then one
different digit; or one digit and then a series of a different digit.
The above series is particularly attractive because the number of threes
in the first seven primes increase by one. There are only eleven other
primes in this series with less then 1800 threes.
See Chris Caldwell JRM 21:4, p 299 &
JRN 22:2, p 101 
Smallest &
Largest Primes
of digit length from 1 to 15
5 & 71
The above two primes evenly divide the sum of the primes less then
themselves.
The only other such prime less then 2,000,000 is 369119, the 1577th prime.
D. Wells, Curious & Interesting
Numbers P.129
Fortunate Primes
Starting with 2 , find product of consecutive primes. Call it
p
Then p + 1 = s
Take next largest prime > s. Call it
v. Then v – p = prime.
Martin Gardner (The Last Recreations) calls these last numbers ‘fortunate
primes’.
Primes in
First k Digits of p
For k = 1,2,6, & 38. The next prime has at least 500 digits !
5, 7, 9, digit Primes
Three consecutive primes sum to a palindromic prime.
Visit Patrick De Geest's very attractive and informative WWW site about
Palindromic Numbers at Other Links
The 9digit set was reported by Jud McCranie July 11, 1998
21 Consecutive Primes
The 21 consecutive primes from 7 to 89 sum to the prime number 953.
Also when arranged in groups of three, each group sums to a prime.
Furthermore, the reverse of these prime sums also sum to 953 !
T.V.Padmakrumar, JRM 27:1, 1995,
p57
Palindromic Primes
11 is the only Palindromic prime with an even number of digits.
These are the smallest and largest Palindromic primes of length 1 to 19.
Number of palindromic primes of length 1=4, 2=1, 3=15, 5=93, 7=668, 9=5172
From PALPRI..HTM by Patrick De
Geese, Belguim, July/96
Four Palindromic
Primes
From PALPRI..HTM byPatrick DeGeese,
Belguim, July/96 Link to his page from Other Links
Palindrome from Primes
Product of the first eight primes divided by ten gives a palindrome
prine number.
9 digit
Palindromic Primes
Palindromic Primes 
There are a total of 5172 nine digit primes that
read the same forward or backward. Many of them have extra
properties. 
Plateau Primes 
There are 3 primes where all the interior digits are
alike and are higher then the terminal digits. There are two primes,
322222223 & 722222227 in which the interior digits are smaller then
the end ones. These are called Plateau and Depression
Primes 
Undulating Primes 
So called when adjacent digits are alternately
greater or less then their neighbors. If there are only two distinct
digits, they are called smoothly undulating. Of the total of 1006
undulating nine digit palindromic primes, seven are smoothly
undulating. 
Peak & Valley Primes 
If the digits of the prime, reading left to right,
steadily increase to a maximum value, and then steadily decrease,
they are called peak primes. Valley primes are just
the opposite. There are a total of 10 peak and 20 valley primes.
345676543 is unique because of the five consecutive digits. 
Les Card JRM 14:1 p30
Depression Primes

The above numbers are called depression primes. The
next ones in the 'two' series contain 27 and 63 two's! Note the
'seven 'two's in the one above. The next ones in the 'five' series
contain 19, 21, 57, 73 & 81 fives. 
Pandigital
Palindromic Primes

This 19 digit number reads the same
forwards and backwards. It contains each of the digits 0 to 9 twice,
except the 7 which appears only once . 
Perfect Prime Squares

In each of these two squares, all rows, columns and
the two main diagonals are distinct prime numbers when read in
either direction. The order5 square above is one of three reported
by Mr. Card.
L. E. Card,Patterns in
Primes, JRM 1:2, 1968, pp .9399, 
Order6
Perfect Prime Squares

In 1998 Carlos B. Rivera and Jaime Ayala
rediscovered the order4 shown above (L. E. Card) and conjecture
that it is the only solution with 20 distinct primes and no
palindromes. They also found another three order5 Perfect Prime
Squares with 24 distinct 5 digit primes (they call them
Primemagical squares). They also found these two order6 squares
which each contain twentyeight 6 digit primes.
Carlos has a WWW page dealing
with Prime Puzzles & Problems at http://www.sci.net.mx/~crivera/. 
9 digit
Superperfect Prime Square

1 of the 24 possible order3 perfect prime squares
(not counting rotations and reflections. Each row, column, and the
two main diagonals all consist of 3digit primes when read in either
direction. This one is superperfect because the broken
diagonal pairs are also 3digit prime numbers. The 5 can be replaced
with an 8. These are the only order3 Superperfect prime squares.
All order 2 and 3 perfect prime squares contain
palindromes and contain duplicate prime numbers. 
Do only Order3 perfect prime squares
contain palindromic primes? Are there any superperfect prime squares of
order greater then 3?
Charles W. Trigg,Perfect Prime
Squares, JRM 17:2, 198485, pp .9194, 198485
Addendum August 31, 2007
Søren Schandorf
and his associates in Denmark have been working on this problem.
Yesterday I received the list of order5 Perfect Palindromic Prime
Squares (PPPS). And some solutions for the order7. Here I show two
examples from his report.
Each order5
PPPS contains seven 5 digit palindromic primes, and each order7
square contains nine 7 digit palindromic primes.
See their report
on this project at
http://www.chronomatics.dk/sppps5.pdf
Søren also confirmed that the two
order3 SPPPS shown on this page are the only ones of that order.
His group found 182 squares for order5 and an astounding 614,157
for order 7. 

Palindromic Sequences
Each sequence is formed from the one above it by inserting n,
the row number, between all adjacent numbers that add to n. k
is the number of numbers in each sequence. So far all k are
prime numbers. Does this series continue indefinitely?
This pattern is credited to Leo
Moser (Martin Gardner, The Last Recreations, p.199).
Palindrome 373

373 = sum of the squares of the first 5 odd primes.
Also: the sum of five consecutive primes starting with 67.
From Patrick De Geest's
Palindrome numbers WWW site at http://www.worldofnumbers.com/ 
Ascending
Pandigital Prime
This prime contains all the digits from 1 to 9 in order, then repeats
starting from 0.
Two similar primes but using only the nine digits from 1 to 9 are
1234567891 and 1234567891234567891234567891.
David Wells, Curious &
Interesting Numbers, p191
Reversible Primes
These primes are six digit reversable with an imbedded four digit
reversable prime.
For example, the top number of the middle column: following are all prime;
311537, 335117, 735113, 711533, 1153, 3511..In this particular case,
31153, 71153 and 35117 are five digit primes, 11 and 53 are two digit
primes, and two 3’s, the 5 and the 7 are all one digit primes.
There are a total of 4769 reversible prime pairs of six digits.
Les Card, JRM 12:4 ,p 27
Reversible Prime 3911

These numbers
are all primes!!
3911, it's reverse, and both numbers with a 3 on either end or a 9
on either end.
There are a total of 102 reversible prime pairs of four digits.
Les Card JRM:11:1 ,p 9 
Digit
Complementary Prime Pairs
A Diigit Complementary Prime Pair is defined as a pair of prime numbers
in which digits in corresponding positions sum to 10 (or 0). There are 136
four digit pairs.
a. a reversible prime pair
b. the two primes
contain 8 different digits
c. twin primes
d.
both primes contain consecutive digits
e. first member of the pair contains the 4 prime
digits in order
f. each prime contains 3 digits the same
Charles W. Trigg, JRM 22:2, 1990,
p 9597
A Prime Circle
This is an example of a prime circle. Two adjacent numbers,
including the last number and the first number, sum to a prime. In this
particular case all the numbers are 3 digits. This circle is of length
ninety, and is part of a 200 length prime circle found by Charles
Ashbacher, JRM 26:1, 1994, p
63.
A String of Primes

Start at the first digit, or the first digit after
any comma, and read a nine digit prime number.
L. E. Card, JRM 11:1, p.16 
Three digits  all
Prime

The only 3digit numbers such that all arrangements
of their three digits are prime numbers. Also for 113, all 2digit
combinations are prime numbers. 
Prime Factors of
114985

The prime factors of 114985 are 1, 5, 13, 29, 61. 
Prime PPDI's
The above three numbers are all Pluperfect
Digital Invariants, meaning that when each digit of the number is
raised to the power equal to the length of the number, the sum of these
powers is equal to the original number.
i.e. 28116440335967 = 2^{14} + 8^{14} +
1^{14} + 1^{14} + 6^{14} + 4^{14} + 4^{14}
+ 0^{14} + 3^{14} + 3^{14} + 5^{14} + 9^{14}
+ 6^{14} + 7^{14}.
The above three numbers are also PRIME !
They are the only primes among the 79 PPDI’s under length forty.
The first number (the smallest) is the only one of the three that is
pandigital. Also, of the four digits that appear
twice, three appear as adjacent pairs.
The largest number contains three digits that appear four times and three digits
that appear three times.
See Deimel & Jones, JRM 14(2),
1982, pp. 87 to 99 for list of the 79 PPDI’S to order 39.
Consecutive
gaps ... primes
Addendum: January, 2006
Luis Rodriguez advised me of another series of primes separated by 0, 2,
4, 6, ..., 26.
The prime number 484511389338941 will produce a string of 14 prime numbers
with gaps the size of the above digits.
Two Prime Pyramids

All the numbers in these pyramids are primes.
Also...
All the numbers in the first pyramid are reversible primes. All
numbers in the second pyramid except the fourth and sixth ones (8 &
12 digits) are also reversible primes. (The next number in each
sequence is composite).
Les.Card JRM 11:4, 1978, 79
,p 28 
Overlapping Primes

The largest known prime number such that any two
adjacent digits are prime and all these primes are different.
David Wells ,Curious and
Interesting Number, p. 195 
All Primes!

These are the largest possible primes with
this property. The top number of the second column as shown on
page 191 of the credit is a typo, as the number shown is
composite. In the third set, the last digit of the first
number could be a three, as that number is also a prime. The
number ‘1’ here is presumed to be prime, although by
definition it is not. Four other numbers with this property
are 233399339, 29399999, 37337999 & 59393339
Chris K. Caldwell,
Journal of Recreational Mathematics, 19:1, 1987, pp 3033
David Wells, Curious & Interesting Numbers pps 191, 192, 200


First & Last
Columns ... are prime numbers
The First 7 Primes
Unfortunately, these relationships do not hold for the next higher
prime, 19.
A Prime Series
(Term n = Term
n1_{ }times 2 plus 1)
Digital sums
of Prime Pair ...
12 digit
Nearrepunit Primes

These are all the 12digit primes of this type.
There is 1 with 3 digits, 1 with 5 digits, 1 with 6 digits,
2 with 8 digits, and 1 with 9 digits.
Next size after 12 digits is 17 with 2 such prime numbers.
C. Caldwell & H.Dubner JRM
27:1, 1995, p 35 
A Prime Pair
(relatively small)

The two largest known twin primes are 242206083 * 2^{38880}
. plus and
minus 1 with 11713 digits, found by Indlekofer and Ja'rai in
November, 1995. They are also the first known gigantic twin primes
(primes with at least 10,000 digits).
See
http://www.utm.edu/research/primes/lists/top20/twin.html 
Common Factors

Each row, column and main diagonal
has a common factor that is one of
the first 8 prime numbers. 
Circular Primes

A circular prime is a prime number that remains
prime as each
leftmost digit (msd) in turn is moved to the right hand side.
From Patrick De Geest's
Palindrome numbers WWW site (see above) 
Priming the Cube

The only arrangement of 8 consecutive digits
(not counting rotations or reflections) such that
any two adjacent sum to a prime number. 
Pairs equal Prime

In this pattern the sum of each pair of numbers
connected by a line sums to a prime number. The pattern is
symmetrical both horizontally and vertically and uses the
consecutive numbers from 1 to 22.
JRM 261, pp 71 solution by
Eryk Cershen of Redwood City, California; from a problem suggested
by Brian Barwell of Middlesex, England 
