Concatenating Emirpimes

Emirpimes have the property that they remain semiprimes when reversed. Thus 15 = 3 x 5 becomes 51 when the digits are reversed and 51 = 3 x 17 which is also a semiprime. What happens to the emirpimes if it is concatenated with its reverse. Does it remain a semiprime? In the case 15, this leads to 1551 = 3 x 11 x 47 which is a sphenic number and not a semiprime. The first such emirpimes is 94 because 9449 = 11 x 859 is a semiprime. In fact all such emirpimes will have 11 as a factor. Here is Gemini's explanation:

The reason no numbers starting with the digit 2 appear in your output comes down to two absolute mathematical rules regarding divisibility: the behavior of even-length palindromes and the properties of even numbers.

When you concatenate a number that begins with 2 with its exact reverse, you are guaranteeing that the resulting number will have at least three prime factors, which automatically disqualifies it from being a semiprime.

Here is the step-by-step breakdown of exactly why this happens:

1. The Concatenated Number is Always an Even-Length Palindrome

When you take any number string and append its exact reverse (e.g., $n = 26$, $rev = 62$, resulting in 2662), you create a perfect mirror image. Because you are doubling the number of digits, the resulting palindrome will always have an even length.

There is a fundamental rule in number theory: Every palindrome with an even number of digits is divisible by 11. Because your program only evaluates these concatenated mirror-images, every single number evaluated in that final step has 11 as one of its prime factors.

2. The Resulting Number Always Ends in 2

If your original number ($n$) starts with the digit 2, its reverse will naturally end with the digit 2. When you concatenate the original number with its reverse, the very last digit of the new string will be the last digit of the reverse.

For example:

• Original: 215

• Reverse: 512

• Concatenation: 215512

Because the concatenated number ends in 2, it is an even number. This means 2 is definitively another prime factor of the concatenated number.

3. The Factor Count Exceeds the Semiprime Limit

By definition, a semiprime must have exactly two prime factors (e.g., $p_1 \times p_2$).

If you start with a number beginning with 2, we have already established that the final concatenated number is divisible by both 11 and 2. Therefore, the number is a multiple of 22.

For a multiple of 22 to be a semiprime, the number would have to be exactly 22 itself (where 2 and 11 are the only factors). For your program to generate 22, the original number would have to be 2, and the reverse would have to be 2. This fails on multiple fronts:

• 2 is a prime number, not a semiprime.

• Your code explicitly filters out palindromes (n != rev_n), so it would ignore 2 anyway.

Any valid concatenated number your program produces that begins with a 2 will be much larger than 22 (the smallest would be 2442, derived from $n=24$). Therefore, its prime factorization will always look like $2 \times 11 \times x$, containing at least three prime factors.

Broader Implications

Because of these exact same mathematical rules, your output will also never contain numbers starting with:

• 4, 6, or 8: The resulting concatenations would all end in an even number, meaning they are divisible by 2 and 11, resulting in too many factors.

• 5: The resulting concatenation would end in a 5, meaning it is divisible by 5 and 11, again resulting in too many factors.

The only numbers that can mathematically survive this final semiprime check are those starting with 1, 3, 7, or 9.

Here is the list of emirpimes that satisfy in the range up to 40,000 (permalink):

=== Emirpime Concatenation Statistics ===

Range evaluated: 1 to 40000

1. Semiprimes: 9790 (24.47% of the range)

2. Emirpimes: 3450 (8.62% of the range)

3. Successful Emirpimes: 476 (1.19% of the range)

94, 115, 169, 185, 341, 355, 394, 398, 766, 794, 923, 926, 955, 985, 1027, 1046, 1141, 1174, 1195, 1202, 1238, 1247, 1306, 1343, 1366, 1382, 1417, 1418, 1466, 1642, 1658, 1703, 1714, 1754, 1757, 1769, 1843, 1891, 1915, 1927, 1985, 3047, 3062, 3071, 3091, 3097, 3098, 3103, 3151, 3166, 3227, 3317, 3385, 3403, 3473, 3599, 3635, 3646, 3653, 3785, 3817, 3866, 3949, 3959, 3979, 3983, 7051, 7058, 7082, 7094, 7165, 7169, 7183, 7186, 7223, 7226, 7235, 7274, 7291, 7318, 7346, 7354, 7367, 7403, 7522, 7637, 7655, 7663, 7715, 7754, 7795, 7849, 7855, 7981, 9121, 9242, 9259, 9346, 9427, 9569, 9578, 9593, 9599, 9617, 9727, 9755, 9809, 9946, 9953, 9983, 9997, 10033, 10085, 10129, 10195, 10261, 10342, 10345, 10462, 10466, 10483, 10517, 10541, 10546, 10558, 10637, 10673, 10705, 10805, 11006, 11141, 11147, 11249, 11302, 11306, 11363, 11366, 11539, 11563, 11591, 11641, 11651, 11653, 11654, 11714, 11723, 11851, 11893, 11917, 11945, 12001, 12013, 12067, 12127, 12146, 12199, 12202, 12217, 12283, 12317, 12361, 12367, 12439, 12458, 12494, 12509, 12674, 12718, 12751, 12758, 12761, 12851, 12877, 12913, 12937, 13102, 13106, 13133, 13142, 13193, 13273, 13322, 13373, 13391, 13459, 13529, 13637, 13739, 13742, 13787, 13817, 13934, 13993, 14015, 14111, 14171, 14213, 14219, 14242, 14261, 14273, 14291, 14318, 14359, 14486, 14501, 14611, 14642, 14647, 14765, 14786, 14834, 14999, 15094, 15157, 15335, 15362, 15389, 15403, 15623, 15658, 15779, 15781, 15793, 15815, 15863, 15874, 15893, 15989, 16003, 16123, 16129, 16153, 16207, 16265, 16331, 16334, 16367, 16387, 16489, 16507, 16859, 16865, 16909, 16969, 17083, 17218, 17261, 17305, 17311, 17441, 17503, 17533, 17611, 17614, 17662, 17687, 18035, 18155, 18463, 18482, 18547, 18613, 18623, 18686, 18689, 18695, 18794, 18806, 18847, 18937, 19015, 19165, 19217, 19246, 19286, 19297, 19331, 19337, 19354, 19358, 19405, 19445, 19579, 19585, 19607, 19637, 19667, 19757, 19858, 19882, 19931, 19969, 30043, 30143, 30182, 30407, 30413, 30437, 30574, 30605, 30607, 30641, 30662, 30731, 30754, 30755, 30779, 30826, 30854, 30865, 30886, 30913, 30991, 30994, 30995, 31127, 31138, 31282, 31342, 31399, 31453, 31807, 31843, 31853, 31946, 31969, 32066, 32071, 32137, 32138, 32177, 32366, 32455, 32458, 32501, 32593, 32602, 32663, 32666, 32765, 32842, 32845, 32891, 33109, 33253, 33257, 33445, 33458, 33559, 33593, 33737, 33991, 34073, 34154, 34169, 34186, 34187, 34205, 34379, 34382, 34409, 34477, 34571, 34598, 34657, 34723, 34745, 34802, 34831, 34907, 34999, 35078, 35135, 35137, 35147, 35173, 35215, 35263, 35269, 35285, 35471, 35522, 35545, 35567, 35723, 35773, 35779, 35813, 35881, 36029, 36035, 36065, 36139, 36143, 36203, 36227, 36311, 36422, 36427, 36446, 36481, 36503, 36535, 36614, 36626, 36631, 36649, 36679, 36733, 36734, 36745, 36794, 36802, 37006, 37031, 37034, 37099, 37153, 37157, 37231, 37331, 37358, 37399, 37438, 37457, 37477, 37487, 37498, 37514, 37574, 37601, 37739, 37834, 37837, 37931, 37946, 37958, 38174, 38209, 38245, 38401, 38413, 38414, 38477, 38497, 38534, 38585, 38597, 38635, 38719, 38785, 38806, 38951, 39037, 39127, 39142, 39205, 39253, 39265, 39289, 39449, 39518, 39523, 39653, 39797, 39826, 39871, 39898, 39899, 39911, 39931, 39947, 39958, 39959, 39977

Emirprimes in Base 10

I was quite surprised to discover that in this mathematical blog (now over ten years old) I'd never once made reference to the term $ \textbf{emirpimes}$ which is simply a semiprime that remains a semiprime (but a different one) when its digits are reversed. The very first emirprimes is 15:$$ \begin{align} 15 &= 3 \times 5 \\ 51 &= 3 \times 17 \end{align}$$Looking back on my posts however, I found that I had posted about such semiprimes before but had not used the term emirpimes to describe them. Specifically, in a post titled Reversible Sphenic Numbers I'd added an addendum that discusses $ \textbf{reversible semiprimes}$ of a very special type when visualised as rectangles:

The above idea (in reference to the earlier discussion about visualising reversible sphenic numbers as rectangular prisms) can be applied to semiprimes as well. In this case we will be working with two dimensional rectangles. The two factors of the semiprime form the length and width of the associated rectangle. In the range up to one million, reversible semiprimes comprise 6.06% of the total range but there are only 26 reversible semiprimes with the property that their associated areas are equal. See permalink for calculation.

The pairs (up to one million) are:

$$ \begin{align} 14269 &= 19 \times 751 \rightarrow 1540 \dots 96241 = 157 \times 613 \rightarrow 1540 \\15167 &= 29 \times 523 \rightarrow 1104 \dots 76151 = 271 \times 281 \rightarrow 1104\\
16237 &= 13 \times 1249 \rightarrow 2524 \dots 73261 = 61 \times 1201 \rightarrow 2524\\
18449 &= 19 \times 971 \rightarrow 1980 \dots 94481 = 107 \times 883 \rightarrow 1980\\
18977 &= 7 \times 2711 \rightarrow 5436 \dots 77981 = 29 \times 2689 \rightarrow 5436\\
36679 &= 43 \times 853 \rightarrow 1792 \dots 97663 = 127 \times 769 \rightarrow 1792\\
140941 &= 97 \times 1453 \rightarrow 3100 \dots 149041 = 103 \times 1447 \rightarrow 3100\\
150251 &= 347 \times 433 \rightarrow 1560 \dots 152051 = 383 \times 397 \rightarrow 1560\\
196891 &= 401 \times 491\rightarrow 1784 \dots  198691 = 431 \times 461 \rightarrow 1784\\
302363 &= 211 \times 1433 \rightarrow 3288 \dots  363203 = 263 \times 1381 \rightarrow 3288\\
308459 &= 173 \times 1783 \rightarrow 3912 \dots 954803 = 937 \times 1019 \rightarrow 3912\\
319853 &= 317 \times 1009 \rightarrow 2652 \dots 358913 = 379 \times 947 \rightarrow 2652\\
958099 &= 761 \times 1259 \rightarrow 4040 \dots 990859 = 839 \times 1181 \rightarrow 4040\\ \end{align} $$

It can be noted that all these reversible semiprimes are odd. Here is a video based on that blog post just mentioned. from that blog postIt was generated using NotebookLM.

I asked Gemini to write a SageMath program identifying all the emirpimes between 1 and 40000 but it failed miserably and so again I tried Claude and this AI succeeded on its first attempt once again showing its undisputed superiority over Gemini when it comes to coding. I won't show them all here but from 28013 onwards they are as follows (permalink):

 ... 28013, 28043, 28049, 28073, 28079, 28093, 28102, 28103, 28117, 28121, 28129, 28139, 28157, 28159, 28166, 28169, 28207, 28213, 28214, 28235, 28271, 28286, 28295, 28298, 28327, 28331, 28333, 28337, 28339, 28343, 28345, 28346, 28354, 28357, 28361, 28363, 28373, 28394, 28415, 28442, 28445, 28451, 28459, 28471, 28481, 28487, 28498, 28507, 28511, 28531, 28553, 28555, 28562, 28606, 28646, 28651, 28693, 28694, 28705, 28777, 28778, 28781, 28783, 28799, 28801, 28829, 28838, 28846, 28849, 28862, 28883, 28891, 28907, 28913, 28922, 28931, 28943, 28957, 28969, 28978, 28981, 28993, 28999, 30001, 30019, 30021, 30027, 30043, 30055, 30061, 30111, 30143, 30146, 30151, 30157, 30182, 30185, 30202, 30227, 30235, 30257, 30263, 30273, 30274, 30279, 30281, 30289, 30311, 30329, 30335, 30353, 30359, 30377, 30386, 30395, 30397, 30407, 30413, 30417, 30421, 30437, 30451, 30453, 30455, 30461, 30463, 30473, 30477, 30479, 30489, 30499, 30518, 30526, 30542, 30551, 30563, 30571, 30574, 30587, 30599, 30601, 30605, 30607, 30611, 30619, 30629, 30641, 30647, 30662, 30673, 30715, 30722, 30731, 30737, 30746, 30754, 30755, 30779, 30787, 30799, 30802, 30813, 30826, 30827, 30854, 30865, 30867, 30886, 30913, 30917, 30922, 30929, 30934, 30946, 30959, 30963, 30967, 30973, 30991, 30994, 30995, 30999, 31007, 31011, 31015, 31021, 31029, 31049, 31061, 31067, 31082, 31105, 31107, 31109, 31118, 31127, 31129, 31138, 31145, 31157, 31166, 31169, 31187, 31197, 31199, 31201, 31235, 31241, 31261, 31282, 31285, 31294, 31299, 31322, 31342, 31345, 31359, 31361, 31366, 31377, 31389, 31399, 31403, 31411, 31435, 31439, 31453, 31457, 31461, 31462, 31463, 31471, 31474, 31478, 31483, 31499, 31522, 31529, 31534, 31553, 31555, 31571, 31587, 31593, 31594, 31645, 31651, 31661, 31669, 31673, 31681, 31693, 31701, 31709, 31718, 31739, 31753, 31754, 31757, 31778, 31791, 31795, 31802, 31803, 31807, 31821, 31826, 31829, 31831, 31839, 31843, 31853, 31867, 31874, 31881, 31903, 31909, 31917, 31918, 31919, 31921, 31927, 31933, 31937, 31942, 31945, 31946, 31953, 31961, 31969, 31979, 31982, 31989, 32041, 32047, 32066, 32071, 32081, 32101, 32134, 32135, 32137, 32138, 32149, 32161, 32177, 32187, 32197, 32209, 32217, 32221, 32222, 32231, 32243, 32254, 32259, 32267, 32293, 32365, 32366, 32374, 32386, 32434, 32446, 32447, 32455, 32458, 32459, 32477, 32498, 32501, 32506, 32519, 32521, 32534, 32546, 32557, 32567, 32577, 32583, 32593, 32597, 32601, 32602, 32638, 32639, 32641, 32649, 32651, 32659, 32663, 32666, 32667, 32673, 32681, 32709, 32711, 32722, 32735, 32738, 32743, 32753, 32755, 32761, 32765, 32815, 32822, 32834, 32842, 32845, 32863, 32866, 32867, 32873, 32881, 32885, 32891, 32954, 32961, 32974, 32986, 32995, 33001, 33007, 33009, 33017, 33035, 33038, 33047, 33067, 33079, 33081, 33097, 33109, 33121, 33127, 33131, 33146, 33163, 33167, 33169, 33171, 33185, 33221, 33241, 33253, 33257, 33262, 33265, 33266, 33279, 33307, 33322, 33357, 33373, 33379, 33382, 33386, 33389, 33398, 33406, 33443, 33445, 33455, 33458, 33499, 33505, 33515, 33545, 33559, 33571, 33593, 33595, 33622, 33643, 33653, 33658, 33667, 33683, 33685, 33689, 33691, 33707, 33709, 33717, 33729, 33731, 33737, 33743, 33763, 33778, 33779, 33783, 33802, 33806, 33815, 33817, 33821, 33837, 33842, 33853, 33869, 33874, 33877, 33881, 33886, 33907, 33913, 33919, 33962, 33973, 33986, 33991, 34003, 34007, 34009, 34049, 34053, 34054, 34058, 34066, 34073, 34079, 34082, 34087, 34091, 34103, 34106, 34109, 34115, 34135, 34139, 34145, 34154, 34163, 34165, 34169, 34179, 34186, 34187, 34193, 34198, 34199, 34205, 34207, 34223, 34234, 34246, 34247, 34249, 34271, 34289, 34291, 34329, 34331, 34334, 34339, 34345, 34373, 34378, 34379, 34382, 34387, 34399, 34409, 34415, 34418, 34433, 34449, 34462, 34477, 34491, 34509, 34517, 34531, 34535, 34555, 34559, 34571, 34579, 34597, 34598, 34601, 34619, 34621, 34633, 34634, 34642, 34654, 34657, 34681, 34691, 34709, 34718, 34723, 34733, 34735, 34745, 34753, 34754, 34761, 34766, 34769, 34789, 34791, 34793, 34801, 34802, 34805, 34811, 34813, 34829, 34831, 34834, 34835, 34851, 34859, 34862, 34863, 34886, 34889, 34898, 34901, 34907, 34909, 34915, 34931, 34951, 34955, 34971, 34973, 34982, 34987, 34993, 34999, 35011, 35018, 35021, 35043, 35057, 35063, 35077, 35078, 35087, 35102, 35135, 35137, 35147, 35151, 35158, 35161, 35173, 35179, 35191, 35195, 35209, 35215, 35218, 35233, 35263, 35269, 35285, 35293, 35297, 35303, 35314, 35331, 35341, 35366, 35377, 35383, 35389, 35403, 35411, 35413, 35429, 35431, 35439, 35459, 35467, 35471, 35474, 35477, 35481, 35493, 35494, 35499, 35501, 35513, 35515, 35522, 35545, 35549, 35551, 35567, 35582, 35599, 35609, 35611, 35614, 35633, 35654, 35669, 35681, 35689, 35691, 35693, 35699, 35713, 35723, 35727, 35737, 35743, 35749, 35755, 35761, 35762, 35769, 35773, 35779, 35813, 35833, 35842, 35849, 35877, 35881, 35885, 35909, 35913, 35935, 35939, 35941, 35947, 35957, 35961, 35971, 35974, 35978, 35981, 36019, 36021, 36026, 36029, 36035, 36053, 36065, 36071, 36079, 36086, 36089, 36122, 36133, 36139, 36143, 36154, 36173, 36178, 36181, 36194, 36199, 36203, 36227, 36238, 36242, 36253, 36257, 36283, 36287, 36289, 36291, 36298, 36301, 36311, 36327, 36338, 36339, 36349, 36357, 36359, 36362, 36367, 36371, 36382, 36397, 36403, 36409, 36413, 36415, 36422, 36427, 36437, 36446, 36447, 36471, 36481, 36483, 36503, 36506, 36509, 36511, 36514, 36521, 36535, 36581, 36591, 36605, 36611, 36614, 36622, 36623, 36626, 36631, 36647, 36649, 36658, 36659, 36661, 36667, 36679, 36681, 36719, 36733, 36734, 36737, 36745, 36758, 36773, 36794, 36802, 36823, 36826, 36831, 36843, 36845, 36851, 36867, 36878, 36881, 36889, 36903, 36911, 36914, 36937, 36959, 36962, 36969, 36977, 36983, 36987, 37001, 37006, 37007, 37031, 37034, 37041, 37043, 37046, 37067, 37069, 37082, 37099, 37106, 37119, 37131, 37137, 37153, 37157, 37163, 37186, 37193, 37203, 37207, 37227, 37229, 37231, 37234, 37237, 37239, 37241, 37255, 37267, 37274, 37285, 37299, 37322, 37331, 37333, 37343, 37358, 37371, 37381, 37382, 37399, 37402, 37403, 37405, 37419, 37435, 37438, 37439, 37457, 37462, 37477, 37487, 37491, 37498, 37514, 37546, 37574, 37586, 37594, 37601, 37603, 37613, 37623, 37651, 37659, 37669, 37678, 37685, 37709, 37711, 37721, 37727, 37729, 37731, 37739, 37753, 37763, 37771, 37787, 37793, 37795, 37798, 37801, 37817, 37822, 37823, 37834, 37837, 37841, 37885, 37894, 37903, 37909, 37931, 37933, 37939, 37943, 37946, 37949, 37955, 37958, 37959, 38015, 38017, 38021, 38027, 38035, 38071, 38074, 38087, 38089, 38091, 38101, 38111, 38117, 38137, 38138, 38141, 38147, 38174, 38179, 38207, 38209, 38217, 38221, 38242, 38245, 38249, 38257, 38263, 38293, 38314, 38341, 38353, 38362, 38381, 38387, 38389, 38401, 38405, 38407, 38413, 38414, 38429, 38435, 38438, 38441, 38455, 38462, 38463, 38474, 38477, 38487, 38495, 38497, 38498, 38515, 38518, 38523, 38527, 38534, 38585, 38597, 38621, 38633, 38635, 38638, 38659, 38667, 38679, 38689, 38697, 38705, 38717, 38719, 38721, 38733, 38751, 38757, 38758, 38761, 38769, 38774, 38782, 38785, 38789, 38795, 38797, 38806, 38807, 38809, 38819, 38834, 38842, 38849, 38854, 38858, 38881, 38882, 38894, 38911, 38919, 38951, 38954, 38963, 38978, 38981, 38987, 38999, 39002, 39007, 39037, 39049, 39062, 39067, 39082, 39086, 39106, 39127, 39131, 39137, 39142, 39143, 39149, 39154, 39166, 39167, 39173, 39205, 39211, 39223, 39253, 39265, 39281, 39283, 39287, 39289, 39297, 39299, 39331, 39335, 39337, 39374, 39381, 39407, 39418, 39437, 39441, 39449, 39453, 39487, 39489, 39513, 39518, 39523, 39526, 39533, 39539, 39547, 39549, 39554, 39557, 39561, 39571, 39577, 39595, 39613, 39626, 39638, 39647, 39651, 39653, 39682, 39686, 39691, 39722, 39731, 39737, 39743, 39745, 39751, 39773, 39777, 39797, 39801, 39811, 39817, 39826, 39838, 39851, 39871, 39874, 39881, 39898, 39899, 39911, 39926, 39931, 39943, 39946, 39947, 39958, 39959, 39977, 39982

The program also shows runs of two and three consecutive emirpimes. There are numerous runs of two but not many of three. There are only two coming up that are below 40000 and these are: 

• 31461, 31462, 31463
• 31917, 31918, 31919

An interesting and rather special emirpimes is 1469. I came across this number via this tweet by Maths Ed:$$ \begin{align} 1469 &= 13 \times 113 \\ 9641 &= 31 \times 311 \end{align}$$I asked Gemini about this curiosity:

1469 = 13 * 113 is an emirpimes because it is a semiprime and, when reversing its digits, we find that 9641 = 31 * 311 is also a semiprime. However, the digits of the factors of 1469 are the reverse of the digits of 9641. This seems rather special. Are there other such emirpimes or this just a one off curiosity.

This is not just a one-off curiosity, though it is a rare property. You have found a special subset of emirpimes where the "reversal" property holds not just for the final number, but for the multiplication process itself.

The Underlying Mechanism: "Multiplication Without Carrying"

The reason $1469$ behaves this way is that the multiplication of its factors ($13 \times 113$) involves no carried digits.

When you multiply numbers normally, you often "carry" digits to the next column (e.g., in $15 \times 15$, $5 \times 5 = 25$, so you write 5 and carry 2). This carrying process scrambles the digits, breaking the symmetry when you reverse the number.

However, if you multiply two numbers and never generate a value greater than 9 in any column of the calculation, the entire process becomes reversible.

Check the math for 1469:

$$13 \times 113 = (10+3) \times (100+10+3)$$

$$= 1000 + 100 + 30 + 300 + 30 + 9$$

$$= 1000 + 400 + 60 + 9 = 1469$$

All the digits ($1, 4, 6, 9$) land perfectly in their slots without overflowing. Because there is no "carry" to disrupt the pattern, reversing the inputs ($31 \times 311$) simply reverses the digits of the output ($9641$).

Are there others?

Yes, but they are scarce because finding primes with only small digits (1, 2, or 3) to avoid carrying is difficult. Most primes contain digits like 7, 8, or 9, which immediately create carries.

Here are a few other members of this "No-Carry" family:

1. 169

   • Equation: $13 \times 13 = 169$

   • Reverse: $31 \times 31 = 961$

   • Note: This is the smallest example, using the emirp 13 squared.

2. 12769

   • Equation: $113 \times 113 = 12769$

   • Reverse: $311 \times 311 = 96721$

   • Note: Like your example, but squaring 113.

3. 1469 (Your find)

   • Equation: $13 \times 113 = 1469$

   • Reverse: $31 \times 311 = 9641$

4. 13481 (A close cousin)

   • Equation: $13 \times 1037$ (Wait, 1037 is $17 \times 61$, not prime).

   • Correction: We are strictly limited to Emirps (reversible primes).

The List of "Pure" Reversible Emirpimes:

If we require the factors to be Emirps (primes that become different primes when reversed) and allow no carries:

• 169 ($13 \times 13$)

• 1469 ($13 \times 113$)

• 12769 ($113 \times 113$)

There are likely others formed by larger combinations of $1, 0, 2, 3$ based emirps (like $1113 \times 13$), but they grow sparse very quickly. You found one of the very few "perfect" examples using distinct primes!