More On Digit Equations

I've posted before about rendering numbers as digit equations, specifically:

• Forming Equations from the Digits of a Number on the 14th March 2024
• The Number Plate Game on the 29th June 2024
• Forming Digit Equations: A Game on the 5th August 2024

Yesterday the number 27586 caught my attention because it can easily be rendered as a digit equation, viz.:$$27586 \rightarrow 2+7+5=8+6$$but it has the special quality that no digits are repeated. This got me wondering what other five digit numbers have this property. In the range of numbers from 27586 to 40000, there are 371 numbers that satisfy the two criteria:

• number has no repeating digits
• number can be split into two parts and a digit equation formed from the sum of the digits on either side of the equal sign

Here are the numbers (permalink):

27586, 27603, 27630, 27801, 27810, 28019, 28037, 28046, 28064, 28073, 28091, 28109, 28136, 28145, 28147, 28154, 28156, 28163, 28165, 28174, 28190, 28307, 28316, 28349, 28361, 28367, 28370, 28376, 28394, 28406, 28415, 28451, 28459, 28460, 28495, 28514, 28541, 28569, 28596, 28604, 28613, 28631, 28640, 28679, 28697, 28703, 28730, 28901, 28910, 29038, 29047, 29056, 29065, 29074, 29083, 29137, 29146, 29148, 29157, 29164, 29173, 29175, 29184, 29308, 29317, 29368, 29371, 29380, 29386, 29407, 29416, 29461, 29470, 29478, 29487, 29506, 29560, 29605, 29614, 29641, 29650, 29704, 29713, 29731, 29740, 29803, 29830, 30126, 30148, 30159, 30214, 30216, 30241, 30249, 30416, 30418, 30425, 30429, 30452, 30461, 30517, 30519, 30526, 30562, 30571, 30618, 30627, 30645, 30654, 30672, 30681, 30719, 30728, 30746, 30764, 30782, 30791, 30829, 30847, 30856, 30865, 30874, 30892, 30948, 30957, 30975, 30984, 31026, 31048, 31059, 31206, 31260, 31408, 31426, 31462, 31480, 31509, 31527, 31572, 31590, 31628, 31682, 31729, 31756, 31765, 31792, 31857, 31875, 31958, 31967, 31976, 31985, 32014, 32016, 32041, 32049, 32104, 32106, 32140, 32160, 32401, 32409, 32410, 32418, 32481, 32490, 32519, 32546, 32564, 32591, 32647, 32674, 32748, 32784, 32849, 32867, 32876, 32894, 32968, 32986, 34016, 34018, 34025, 34029, 34052, 34061, 34106, 34108, 34126, 34160, 34162, 34180, 34205, 34209, 34218, 34250, 34281, 34290, 34502, 34520, 34601, 34610, 34658, 34685, 34759, 34768, 34786, 34795, 34869, 34896, 35017, 35019, 35026, 35062, 35071, 35107, 35109, 35127, 35170, 35172, 35190, 35206, 35219, 35246, 35260, 35264, 35291, 35602, 35620, 35701, 35710, 35769, 35796, 35879, 35897, 36018, 36027, 36045, 36054, 36072, 36081, 36108, 36128, 36180, 36182, 36207, 36247, 36270, 36274, 36405, 36450, 36458, 36485, 36504, 36540, 36702, 36720, 36801, 36810, 37019, 37028, 37046, 37064, 37082, 37091, 37109, 37129, 37145, 37154, 37156, 37165, 37190, 37192, 37208, 37248, 37280, 37284, 37406, 37415, 37451, 37459, 37460, 37468, 37486, 37495, 37514, 37541, 37569, 37596, 37604, 37640, 37802, 37820, 37901, 37910, 38029, 38047, 38056, 38065, 38074, 38092, 38146, 38157, 38164, 38175, 38209, 38245, 38249, 38254, 38267, 38276, 38290, 38294, 38407, 38416, 38425, 38452, 38461, 38469, 38470, 38496, 38506, 38524, 38542, 38560, 38579, 38597, 38605, 38614, 38641, 38650, 38704, 38740, 38902, 38920, 39048, 39057, 39075, 39084, 39147, 39156, 39158, 39165, 39167, 39174, 39176, 39185, 39246, 39264, 39268, 39286, 39408, 39417, 39426, 39462, 39471, 39480, 39507, 39516, 39561, 39570, 39615, 39624, 39642, 39651, 39705, 39714, 39741, 39750, 39804, 39840

I've looked at the first number in this sequence so let's look at the last:$$39840 \rightarrow 3+9=8+4+0$$The sequence will eventually terminate because the largest number possible will contain all ten digits but what might this number be? For a start it can't contain all the digits from 0 to 9 because the sum of these digits is 45 and can't be divided into two equal parts. So we have to drop the 1 if looking for the largest possible number. I think the largest possible number is 985647320 where we have:$$985764320 \rightarrow 9+8+5=7+6+4+3+2+0$$It's also possible to swap the 5 on the left with the 32 on the right so that we get 983276540 so that we have:$$983276540 \rightarrow 9+8+3+2=7+6+5+4+0$$However, this number is smaller than the previous and so it is not the largest possible. What about five digit numbers that satisfy the following criteria:

• number has no repeating digits
• number can be split into two parts and a digit equation formed from the product of the digits on either side of the equal sign

In this case, between 27586 and 40000, there are only 38 numbers that qualify (permalink):

29136, 29163, 29316, 29361, 29613, 29631, 31426, 31462, 31629, 31692, 31846, 31864, 32649, 32694, 34126, 34162, 34216, 34261, 34612, 34621, 34689, 34698, 36129, 36192, 36219, 36249, 36291, 36294, 36489, 36498, 36912, 36921, 38146, 38164, 38416, 38461, 38614, 38641

Let's take the first number in this list, 29136, where we have:$$29136 \rightarrow 2 \times 9 = 1 \times 3 \times 6$$The last number in the list is 38641 where we have:$$38641 \rightarrow 3 \times 8 = 6 \times 4 \times 1$$Other variations on this theme are possible such as using the sum of squares of the digits. For example, let's propose the criteria:

• number has no repeating digits
• number can be split into two parts and a digit equation formed from the sum of the squares of the digits on either side of the equal sign

These criteria yield 48 numbers in the range from 27586 to 40000. These are (permalink):

27614, 27641, 27658, 27685, 27869, 27896, 28769, 28796, 29067, 29076, 29607, 29670, 29706, 29760, 30267, 30627, 31857, 31875, 32067, 32607, 32670, 34517, 34571, 35417, 35471, 36027, 36207, 36245, 36254, 36270, 36425, 36452, 36524, 36542, 38157, 38175, 39158, 39185, 39457, 39475, 39518, 39547, 39574, 39581, 39745, 39754, 39815, 39851

Let's take the first number in the list, 27614, as an example:$$27614 \rightarrow 2^2+7^2 = 6^2+1^2+4^2$$The last number in the list, 39851, can be split as follows:$$39851 \rightarrow 3^2+9^2=8^2+5^2+1^2$$More variations are possible of course but that will do for now. Forming digit equations from the digits of a number falls most definitely into the realm of recreational mathematics and is base-10 specific but it's an interesting mental exercise and what's wrong with having fun with numbers anyway.

These types of mental exercises, especially for children, can serve as an easy entry point to number theory after which they can embark on a deeper exploration of number properties that are not base-specific and that are intrinsic to the number itself, such as primeness and the sum of a number's divisors relative to the number itself that determines whether it is deficient, perfect or abundant.