Some Properties of the Number 2023

For my one hundredth post of the year 2022, I thought I'd create a post about the forthcoming year before it arrives. Firstly, looking at Numbers Aplenty, we find that a 4×4 square can be tiled in 2023 ways using the L triomino and the monomino. Figure 1 shows four examples and click here to see all 2023 possible arrangements. See OEIS A220054 and A127870.

Figure 1

Another interesting property that Numbers Aplenty brings up is the following:

2023 = (2+0+2+3)(2²+0²+2²+3²)²

It also turns out that the prime factors, when concatenated, form a palindrome.

2023 = 7 x 17 x 17 --> 71717

This as it turns out is not a common property (see OEIS A046447). In the range of composite numbers up to 40,000, there are only 193 such numbers. In terms of years, it has been 320 years since this last happened and next occurrence will be rather sooner, in 2048. Here are the numbers (permalink):

4, 8, 9, 16, 25, 27, 32, 39, 49, 64, 69, 81, 119, 121, 125, 128, 129, 159, 219, 243, 249, 256, 259, 329, 339, 343, 403, 429, 469, 507, 512, 625, 669, 679, 729, 795, 1024, 1207, 1309, 1329, 1331, 1533, 1547, 1587, 1589, 1703, 2023, 2048, 2097, 2187, 2319, 2321, 2359, 2401, 2649, 2701, 3039, 3125, 3421, 3503, 3629, 3633, 3639, 3729, 3897, 3899, 3975, 4096, 4303, 4607, 4719, 4839, 5289, 5295, 5547, 5565, 5603, 5739, 6209, 6401, 6561, 6591, 6597, 6979, 7269, 7663, 7819, 7833, 8029, 8192, 8203, 8427, 8519, 8745, 8921, 9329, 9583, 9703, 9707, 9795, 10001, 10021, 10201, 10299, 10599, 10731, 11199, 11499, 12207, 12729, 12795, 13129, 13489, 13573, 13801, 13929, 14133, 14147, 14189, 14295, 14399, 14641, 14829, 15459, 15463, 15549, 15563, 15625, 15987, 16233, 16384, 16507, 16807, 16863, 16959, 17161, 17295, 17363, 17859, 18489, 18789, 18829, 19003, 19683, 19689, 19875, 19907, 20111, 20289, 20489, 20589, 20667, 21259, 21959, 22139, 22507, 22719, 22801, 23019, 23619, 24601, 24901, 25459, 26349, 27879, 28273, 28601, 28609, 28847, 28951, 29729, 29829, 31129, 31363, 31423, 31829, 32357, 32761, 32768, 33173, 33339, 34391, 35143, 36481, 36501, 36763, 36849, 38307, 38899, 38955, 39599, 39939, 39963

On the topic of palindromes, the sum of 2023 and its reverse 3202 is a palindrome. Of all the numbers from 1 to 40,000, there are 9456 (or almost 25%) with this property.

2023 + 3202 = 5225

2023 is also palindromic in base 16 since it equals 7e7. In the range of numbers up to 40,000, this occurs 409 times.

2023 = 7e7

The numbers are:

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 34, 51, 68, 85, 102, 119, 136, 153, 170, 187, 204, 221, 238, 255, 257, 273, 289, 305, 321, 337, 353, 369, 385, 401, 417, 433, 449, 465, 481, 497, 514, 530, 546, 562, 578, 594, 610, 626, 642, 658, 674, 690, 706, 722, 738, 754, 771, 787, 803, 819, 835, 851, 867, 883, 899, 915, 931, 947, 963, 979, 995, 1011, 1028, 1044, 1060, 1076, 1092, 1108, 1124, 1140, 1156, 1172, 1188, 1204, 1220, 1236, 1252, 1268, 1285, 1301, 1317, 1333, 1349, 1365, 1381, 1397, 1413, 1429, 1445, 1461, 1477, 1493, 1509, 1525, 1542, 1558, 1574, 1590, 1606, 1622, 1638, 1654, 1670, 1686, 1702, 1718, 1734, 1750, 1766, 1782, 1799, 1815, 1831, 1847, 1863, 1879, 1895, 1911, 1927, 1943, 1959, 1975, 1991, 2007, 2023, 2039, 2056, 2072, 2088, 2104, 2120, 2136, 2152, 2168, 2184, 2200, 2216, 2232, 2248, 2264, 2280, 2296, 2313, 2329, 2345, 2361, 2377, 2393, 2409, 2425, 2441, 2457, 2473, 2489, 2505, 2521, 2537, 2553, 2570, 2586, 2602, 2618, 2634, 2650, 2666, 2682, 2698, 2714, 2730, 2746, 2762, 2778, 2794, 2810, 2827, 2843, 2859, 2875, 2891, 2907, 2923, 2939, 2955, 2971, 2987, 3003, 3019, 3035, 3051, 3067, 3084, 3100, 3116, 3132, 3148, 3164, 3180, 3196, 3212, 3228, 3244, 3260, 3276, 3292, 3308, 3324, 3341, 3357, 3373, 3389, 3405, 3421, 3437, 3453, 3469, 3485, 3501, 3517, 3533, 3549, 3565, 3581, 3598, 3614, 3630, 3646, 3662, 3678, 3694, 3710, 3726, 3742, 3758, 3774, 3790, 3806, 3822, 3838, 3855, 3871, 3887, 3903, 3919, 3935, 3951, 3967, 3983, 3999, 4015, 4031, 4047, 4063, 4079, 4095, 4097, 4369, 4641, 4913, 5185, 5457, 5729, 6001, 6273, 6545, 6817, 7089, 7361, 7633, 7905, 8177, 8194, 8466, 8738, 9010, 9282, 9554, 9826, 10098, 10370, 10642, 10914, 11186, 11458, 11730, 12002, 12274, 12291, 12563, 12835, 13107, 13379, 13651, 13923, 14195, 14467, 14739, 15011, 15283, 15555, 15827, 16099, 16371, 16388, 16660, 16932, 17204, 17476, 17748, 18020, 18292, 18564, 18836, 19108, 19380, 19652, 19924, 20196, 20468, 20485, 20757, 21029, 21301, 21573, 21845, 22117, 22389, 22661, 22933, 23205, 23477, 23749, 24021, 24293, 24565, 24582, 24854, 25126, 25398, 25670, 25942, 26214, 26486, 26758, 27030, 27302, 27574, 27846, 28118, 28390, 28662, 28679, 28951, 29223, 29495, 29767, 30039, 30311, 30583, 30855, 31127, 31399, 31671, 31943, 32215, 32487, 32759, 32776, 33048, 33320, 33592, 33864, 34136, 34408, 34680, 34952, 35224, 35496, 35768, 36040, 36312, 36584, 36856, 36873, 37145, 37417, 37689, 37961, 38233, 38505, 38777, 39049, 39321, 39593, 39865

2023 is also a:

• Duffinian number: a composite number that has no factors in common with its sum of divisors \( \sigma(\tau)) \).
  
  
• Harshad number: a number that is divisible by the sum of its digits. In fact, 2022 was a Harshad number and so are 2024 and 2025 we have a run of four consecutive Harshad numbers.
  
  
• Lucky number: a number that survives a sieving process which is similar to the Eratosthenes sieve that can be used to compute the prime numbers.
  
  
• Junction number: a number that can be formed in two ways by adding a number to its sum of digits. Here we have 1997 + sod(1997) = 2015 + sod(2015) = 2023 where sod means sum of digits.

Turning to the Online Encyclopedia of Integer Sequences (OEIS), we find that it is a member of OEIS A063416:

A063416

Multiples of 7 whose sum of digits is equal to 7.

The sequence begins:

7, 70, 133, 322, 511, 700, 1015, 1141, 1204, 1330, 2023, 2212, 2401, 3031, 3220, 4102, 5110, 7000, 10024, 10150, 10213, 10402, 11032, 11221, 11410, 12040, 12103, 13111, 13300, 15001, 20041, 20104, 20230, 21112, 21301, 22120, 23002, 24010 

There is a strong connection between 2023 and the digit 7:

• the sum of its digits is 7
• it is divisible by 7
• both of its factors (7 and 17) contain the digit 7
• in octal, it has the representation 3747
• in hexadecimal, it has the representation 7e7
• in base 17, it has the representation 700
• its Fibonacci seed numbers are 17 and 49 = 7 x 7

By "Fibonacci seed numbers" I mean that by starting with 17 and 49 and proceeding Fibonacci-style, we get to 2023 thus:

17, 49, 66, 115, 181, 296, 477, 773, 1250, 2023

It is a member of OEIS A070258:

A070258

Smallest of 3 consecutive numbers each divisible by a square.

• \(2023 = 7 \times 17^2\)
• \(2024 = 2^3 \times 11 \times 23\)
• \(2025 = 3^4 \times 5^2\)

The initial members of the sequence are:

48, 98, 124, 242, 243, 342, 350, 423, 475, 548, 603, 724, 774, 844, 845, 846, 1024, 1250, 1274, 1323, 1375, 1420, 1448, 1519, 1664, 1674, 1680, 1681, 1682, 1848, 1862, 1924, 2007, 2023, 2056, 2106, 2150, 2223, 2275, 2348, 2366, 2523, 2527, 2574, 2644

Well, that's largely what I decided to present from Numbers Aplenty and the OEIS. What else is there of interest about the number 2023? I thought I'd ask ChatGPT the following question: what are some interesting facts, mathematical or otherwise, about the number 2023? Figure 2 is a screenshot of what it spat out.

Figure 2

As usual with ChatGPT, when it comes to Mathematics, it can spew out some nonsense. In case, it claims 2023 is a prime number! Similarly, it's claim that "2023 is the sum of three consecutive prime numbers: 661 + 673 + 677 is false. These are indeed consecutive prime numbers but they add up to 2011. Can it get any worse? Well, yes. The claim is made that "2023 is the hypotenuse of a Pythagorean triple: (683, 992, 2023)" and testing this out we find that \(683^2+992^2=1,450,553\) while \(2023^2=4,092,529\).  We're not done yet! Another bold claim is made that "2023 can be written as the sum of two cubes in two different ways: \(2023 = 13^3 + 2^3 = 11^3 + 10^3\) ". Nope. \(13^3+2^3=2205\) and \(11^3+10^3=2331\). It turns out that 2023 cannot be expressed as a sum of two cubes. A final claim is made that 2023 has the representation 11111011111 in binary. No. The correct representation is 11111100111.

Clearly time to give up on ChatGPT as a source of information about Mathematics. I've written about this AI bot's deficiencies in a post titled ChatGPT and a Tragi-Comedy of Integration. Let's not leave off without a final laugh. After the bot's spectacular failures, I asked it: 

I'm impressed in a negative way. Except for points 2 and 6, every mathematical statement that you made was  wrong. 2023 is not a prime number and so on. How can you get things so wrong? 

Its response was: 

I apologize for any errors in my previous response. As a language model, I can provide information and answer questions to the best of my ability based on the knowledge and information that I have been trained on. However, I am not infallible and it is possible for me to make mistakes or provide incorrect information.

In fact, 2023 is not a prime number. It can be divided evenly by 3, 11, and 61, so it is a composite number. I apologize for any confusion this may have caused.

I hope this clears up any misunderstandings and I apologize for any errors in my previous response. Please feel free to ask me any other questions you may have.

Actually, looking at the atomic number of Calcium, I checked and found that it was 20 not 2023. This perhaps gives an insight into what's going wrong with the bot. It's looking at the first two digits (20) and ignoring the last two digits (23). As for the Mathematics, I simply don't know how its coming up with its answers.

All triprimes can be envisaged as "bricks", 3-D solid objects. If all three primes are distinct, we have a sphenic number and an associated sphenic brick. The volume of the brick is the number itself but there is always an associated number: the surface area of the brick. In the case of 2023, the associated number is 1054 (square units). This number is sphenic and the associated sphenic brick has a surface area of 1246 square units. This number in turn is sphenic and the associated area is 1630 square units. Once again, 1630 is sphenic with associated surface area of 2302 square units. At this point, the chain is broken because 2302 is biprime. This is summarised below:$$ \begin{align} 2023 &= 7 \times 17 \times 17 &\rightarrow 1054\\1054 &= 2 \times 17 \times 31 &\rightarrow 1246\\1246 &= 2 \times 7 \times 89 &\rightarrow 1630\\1630&= 2 \times 5 \times 163 &\rightarrow 2302\\2302 &= 2 \times 1151 \end{align}$$Figure 3 shows a list of the triprime numbers between 2000 and 2100. Thus there is another twelve years before another triprime number, 2035, is encountered. 2035 is also a sphenic number because it has three distinct factors

Figure 3