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
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:
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:
169
Equation: $13 \times 13 = 169$
Reverse: $31 \times 31 = 961$
Note: This is the smallest example, using the emirp 13 squared.
12769
Equation: $113 \times 113 = 12769$
Reverse: $311 \times 311 = 96721$
Note: Like your example, but squaring 113.
1469 (Your find)
Equation: $13 \times 113 = 1469$
Reverse: $31 \times 311 = 9641$
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!
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