## Using all digits from 1 to 9 to express Palindromes < 1000

Assign a different digit from 1 to 9 to each letter in the expression

$(A-B)^c \; + \; (C-D)^E \; + (F-G)^I$   a palindrome < 1000

Paul found that there no solutions for that format to equal the following numbers

330,350,366,394,399,
414,429,434,437,438,441,444,461,463,554,
712,
810,821,823,827,828,831,834,835,836,838,868,881,
910,914,916,918,920,922,923,925,926,927,929,940,948,997

In that list, we note these palindromes:   414, 434, 444, 828, 838, 868, 929

So there are solutions for the other palindromes   < 1000

1,2,3,4,5,6,7,8,9,
11,22,33,44,55,66,77,88,99,
101,111,121,131,141,151,161,171,181,191,
202,212,222,232,242,252,262,272,282,292,
303,313,323,333,343,353,363,373,383,393,
404,424,454,464,474,484,494,
505,515,525,535,545,555,565,575,585,595,
606,616,626,636,646,656,666,676,686,696,
707,717,727,737,747,757,767,777,787,797,
808,818,828,848,858,878,888,898,
909,919,939,949,959,969,979,989,999

Verify whether the 5764675 is the largest palindrome that can be obtained in this fashion

$(4-6)^7 \; + \; (7-5)^1 \; + \; (2-9)^8 \; = \; 5764675$
$(4-6)^7 \; + \; (7-5)^1 \; + \; (9-2)^8 \; = \; 5764675$
$(6-4)^1 \; + \; (1-3)^7 \; + \; (2-9)^8 \; = \; 5764675$
$(2-9)^8 \; + \; (8-6)^1 \; + \; (3-5)^7 \; = \; 5764675$
$(6-4)^1 \; + \; (1-3)^7 \; + \; (9-2)^8 \; = \; 5764675$
$(4-5)^1 \; + \; (1-6)^3 \; + \; (9-2)^8 \; = \; 5764675$
$(4-5)^1 \; + \; (1-6)^3 \; + \; (2-9)^8 \; = \; 5764675$
$(9-2)^8 \; + \; (8-6)^1 \; + \; (3-5)^7 \; = \; 5764675$
$(1-6)^3 \; + \; (3-4)^5 \; + \; (2-9)^8 \; = \; 5764675$
$(4-5)^9 \; + \; (9-2)^8 \; + \; (1-6)^3 \; = \; 5764675$
$(1-6)^3 \; + \; (3-4)^7 \; + \; (9-2)^8 \; = \; 5764675$
$(1-6)^3 \; + \; (3-4)^7 \; + \; (2-9)^8 \; = \; 5764675$
$(1-6)^3 \; + \; (3-4)^5 \; + \; (9-2)^8 \; = \; 5764675$

Paul confirmed that 5764675 is the largest palindrome that can be expressed in this manner.

Paul found these palindromes can also be expressed in this manner:

4-digit palindromes:

{1001,1111,1331,1441,1551,1661,1881,2222,2332,2442,2662,2772,2992,3003,3113,3443,3553,
3883,3993,4004,4114,4224,4334,5115,5665,6446,6556,6776,7337,7557,7667,7777,8448,8998}

5-digit palindromes:

{11911,13431,14341,15651,15751,16261,16361,16461,16561,16861,17071,17371,18781,19491,
19591,19691,20902,22122,22222,23432,32923,33033,33433,34443,36063,36463,36863,39393,
43534,44344,45854,46464,46664,46764,47174,47274,47674,52825,58385,58685,58785,63036,
63436,65056,65556,65656,65756,65856,66166,66266,66566,68786,69696,73337,73937,77677,
77877,78087,78187,78387,79097,79197,81918,82228}

6-digit palindromes:

{117711,259952,262262,277772,279972,308803}

7-digit palindromes:

{1679761,1726271,5761675,5764675}

The largest prime numbers:

$(8 - 1)^9 \; + \; (9 - 3)^7 \; + \; (4 - 6)^2 \; = \; 40633547$
$(8 - 1)^9 \; + \; (9 - 3)^7 \; + \; (6 - 4)^2 \; = \; 40633547$

These are palindromes which are the product of two consecutive primes, can be expressed in this manner.
$6 \; = \; 2 \; \times \; 3$
$77 \; = \; 7 \; \times \; 11$
$323 \; = \; 17 \; \times \; 19$
$36863 \; = \; 191 \; \times \; 193$

1115111 cannot

$1115111 \; = \; 1051 \; \times \; 1061$

math grad - Interest: Number theory
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### 4 Responses to Using all digits from 1 to 9 to express Palindromes < 1000

1. paul says:

Here are all the Palindromic solutions, indicating that 5764675 is the largest.

{1,2,3,4,5,6,7,8,9,11,22,33,44,55,66,77,88,99,101,111,121,131,141,151,161,171,181,191,202,212,222,232,242,252,262,272,282,292,303,313,323,333,343,353,363,373,383,393,404,424,454,464,474,484,494,505,515,525,535,545,555,565,575,585,595,606,616,626,636,646,656,666,676,686,696,707,717,727,737,747,757,767,777,787,797,808,818,848,858,878,888,898,909,919,939,949,959,969,979,989,999,1001,1111,1331,1441,1551,1661,1881,2222,2332,2442,2662,2772,2992,3003,3113,3443,3553,3883,3993,4004,4114,4224,4334,5115,5665,6446,6556,6776,7337,7557,7667,7777,8448,8998,11911,13431,14341,15651,15751,16261,16361,16461,16561,16861,17071,17371,18781,19491,19591,19691,20902,22122,22222,23432,32923,33033,33433,34443,36063,36463,36863,39393,43534,44344,45854,46464,46664,46764,47174,47274,47674,52825,58385,58685,58785,63036,63436,65056,65556,65656,65756,65856,66166,66266,66566,68786,69696,73337,73937,77677,77877,78087,78187,78387,79097,79197,81918,82228,117711,259952,262262,277772,279972,308803,1679761,1726271,5761675,5764675}

Paul.

2. paul says:

These are the 2 largest resulting Primes

(8 – 1)^9 + (9 – 3)^7 + (4 – 6)^2 = 40633547
(8 – 1)^9 + (9 – 3)^7 + (6 – 4)^2 = 40633547
P.

• benvitalis says:

Cool! Thanks for exploring further in this puzzle

• benvitalis says:

6, 77, 323, 36863 are palindromes which are the product of two consecutive primes can be expressed in this manner. 1115111 (a 7-digit palindrome) cannot
$1115111 \; = \; 1051 \; \times \; 1061$