Fermat Pseudoprimes

Today I turned 25351 days old and the number 25351 turns out to have some quite interesting properties connected with pseudoprimes. To quote from Wikipedia:

In number theory, the Fermat pseudoprimes make up the most important class of pseudoprimes that come from Fermat's little theorem. Fermat's little theorem states that if p is prime and a is coprime to p, then \(a^{p−1} − 1 \) is divisible by p. 

For an integer a > 1, if a composite integer x divides \( a^{x−1} − 1, \) then x is called a Fermat pseudoprime to base a. In other words, a composite integer is a Fermat pseudoprime to base a if it successfully passes the Fermat primality test for the base a. It follows that if x is a Fermat pseudoprime to base a, then x is coprime to a. 

The smallest base-2 Fermat pseudoprime is 341. It is not a prime, since it equals 11·31, but it satisfies Fermat's little theorem: \( 2^{340} ≡ 1 \) (mod 341) and thus passes the Fermat primality test for the base 2. Pseudoprimes to base 2 are sometimes called Poulet numbers, after the Belgian mathematician Paul Poulet, Sarrus numbers, or Fermatians (sequence A001567 in the OEIS). 

A Fermat pseudoprime is often called a pseudoprime, with the modifier Fermat being understood. An integer x that is a Fermat pseudoprime for all values of a that are coprime to x is called a Carmichael number.

Now 25351 is a member of OEIS A005936: Pseudoprimes to base 5. This is because \( 5^{25350} ≡ 1 \) (mod 25351). However, it turns out that 25351 is not simply a Fermat pseudoprime. It is also a strong pseudoprime defined by Wikipedia as:

In number theory, a probable prime is a number that passes a primality test. A strong probable prime is a number that passes a strong version of a primality test. All primes pass these tests, but a small fraction of composites also pass, making them "false primes". Unlike the Fermat pseudoprimes, for which there exist numbers that are pseudoprimes to all coprime bases (the Carmichael numbers), there are no composites that are strong pseudoprimes to all bases.

Formally, an odd composite number \(n = d \times 2^s + 1 \) with d also odd is called a strong (Fermat) pseudoprime to a base a when one of the following conditions holds:$$ a^{d} \equiv 1\! \mod n \text{ or}$$$$a^{d\cdot 2^{r}}\equiv -1 \!\mod n\quad {\mbox{ for some }}0\leq r < s $$ If a number n satisfies one of the above conditions and we don't yet know whether it is prime, it is more precise to refer to it as a strong probable prime to base a. But if we know that n is not prime, then one may use the term strong pseudoprime. The definition is trivially met if a ≡ ±1 mod n so these trivial bases are often excluded. Guy mistakenly gives a definition with only the first condition, which is not satisfied by all primes.

Now 25351 = 2 * 12675 + 1 satisfies the \(n = 12675\times 2^s + 1 \) form and 

\( 5^{12675} ≡ 1 \) (mod 25351). Hence 25351 is a strong pseudoprime. This condition may seem somewhat arbitrary but there is good explanation of why it's so at WolframAlpha. 

Things get more complicated still when we learn that 25351 is also a Euler pseudoprime but I'm not going to delve any deeper at the moment. That's probably enough except that I'll finish up by saying a little about the Carmichael numbers that satisfy Fermat's little theorem to all coprime bases. The first of these numbers are 561, 1105, 1729, 2465, 2821, 6601, 8911, 10585, 15841 and 29341. Carmichael numbers have at least three positive prime factors e.g. 561 = 3 * 11 * 17.

ADDENDUM: entered on March 11th 2020

I'm revisiting pseudoprimes because it's a topic about which I always seem to get confused. I've mentioned them previously in the following posts:

• Fermat Pseudoprimes on the 30th August 2018
• Carmichael Numbers on the 30th January 2019  
• The Original Taxi Cab Number in a New Light on the 21st December 2019

What brought to mind again was the number 25909, the number of days old that I was yesterday. One of its properties is that it's a member of OEIS A020267: strong pseudoprimes to base 41. Now it's easy enough to show that it's a pseudoprime to base 41 because$$41^{25908}-1 \equiv 0 \bmod{25909}$$However, to show 25909 is a strong pseudoprime, it needs to fulfil one of two possible conditions. Formally, an odd composite number \(n = d \times 2^s + 1\) with d also odd is called a strong (Fermat) pseudoprime to a base, when one of the following conditions holds:$$ a^{d} \equiv 1\! \mod n \text{ or}$$$$a^{d\cdot 2^{r}}\equiv -1 \!\mod n\quad {\mbox{ for some }}0\leq r \leq s $$So in the case of 25909 we have: $$25909=25908+1=2^2 \times 3 \times 17 \times 127+1=6477 \times 2^2+1$$The value of \(d\) here is 6477 so let's test it out using the first condition. We find that:$$41^{6477} \equiv 8805 \bmod{25909}$$Clearly it does not pass the first test, so let's try the second.$$41^{6477*2} \equiv -1 \bmod 25909$$It passes and so 25909 can claim to be a strong pseudoprime.

Quadratic Reciprocity

I just read a most interesting article in Quanta Magazine about the new Fields medallist Peter Scholze. His discoveries are quite beyond my comprehension but in the article mention is made of quadratic reciprocity. I've come across the concept repeatedly but have always avoided taking the time to understand it. However, in this article it was explained simply and clearly. 

Reciprocity laws are generalisations of the 200-year-old quadratic reciprocity law, a cornerstone of number theory and one of Scholze’s personal favourite theorems. The law states that given two prime numbers p and q, in most cases p is a perfect square on a clock with q hours exactly when q is a perfect square on a clock with p hours. For example, five is a perfect square on a clock with 11 hours, since \(5 = 16 = 4^2 \), and 11 is a perfect square on a clock with five hours, since \(11 = 1 = 1^2 \).

Here is a more detailed explanation of quadratic reciprocity (source):

It's easy to find integer solutions of the equation \( y^2 = 2x^2 + 7k \), but there are no integer solutions at all of the seemingly similar equation \( y^2 = 3x^2 + 7k \).  This is the sort of thing that experienced number theorists can tell at a glance, simply by noting that the equations written modulo 7 are \( y^2 = 2x^2 \) and \( y^2 = 3x^2 \) respectively, and since division is unique in the field of integers (mod p) we have \( (y/x)^2 \equiv n \pmod{p} \), which implies that n must be the square of some element of the field of integers modulo p.  But the integers mod 7 are just 0, 1, 2, 3, 4, 5, 6, whose squares (mod 7) are 0, 1, 4, 2, 2, 4, 1. Thus the equation \( y^2 = nx^2 + 7k \) can have integer solutions only if n is congruent to 0, 1, 2, or 4 (mod 7).  These are called the quadratic residues (mod 7), and the remaining numbers 3, 5, 6 are called the non-quadratic residues.  

It's often extremely important when dealing with problems in number theory to know whether a certain prime p is a square (i.e., a quadratic residue) modulo some other particular prime q. Legendre defined a symbol to represent this information, which we will denote as

This is a remarkable fact, and not at all self-evident.  Legendre succeeded in proving some special cases of this, and also gave what he thought was a complete proof, but his argument relied on the premise that every arithmetic progression contains infinitely many primes. This is in fact true, as subsequently shown by Dirichlet, but at the time of Legendre's proof it wasn't known, so Gauss pointed out that Legendre's proof was incomplete. Gauss went even further, and gave several (valid) proofs of this remarkable theorem, which he called the Fundamental Theorem of number theory.

Thus [p\q] equals either +1 or -1 depending on whether p is or isn't a square modulo q. From examining many individual cases, Euler had previously noticed a striking relationship between the [p\q] and [q\p].  Specifically, he noticed that [p\q] = [q\p] except when p and q are both of the form 4k-1, in which case [p\q] = -[q\p]. 

Let's take a specific example. Yesterday I turned 25339 days old. The previous prime is 25321. It can seen that \( 25339 \equiv 18 \pmod{25321} \) and \( 25321 \equiv 25321 \pmod{253339} \), neither residue being square. Therefore \([25339\25321] = [25321\25339] = -1\). Note that 25321 is a 4k+1 prime while 25339 is a 4k-1 prime, which is in accord with the theorem.

I've download a PDF titled Quadratic Residues and Non-Residues Selected Topics by Steve Wright from the Department of Mathematics and Statistics Oakland University, Rochester, Michigan and dated 21st October 2106. It's freely available here. I'll have a read over this and see if I can deepen my understanding of this topic.

Here are some SAGE commands that can be used when dealing with quadratic residues (double click to enlarge):

Source: http://doc.sagemath.org/html/en/constructions/number_theory.html

ADDENDUM:

When trying to tackle the above Quadratic Residues and Non-Residues Selected Topics, I quickly realised that my foundations in modular arithmetic were a little shaky and needed to be reinforced a little. I began watching a series of videos by Polar Pi on YouTube, beginning with this video on a linear congruence equation with one solution:

The next video in this series relates to linear congruence equations with multiple solutions and the third of the three videos looks at linear congruence equations with no solutions.

The YouTube channel blackpenredpen has some videos on modular arithmetic as well:

• What does a ≡ b (mod n) mean? Basic Modular Arithmetic, Congruence? Link
• Solving congruences, 3 introductory examples. Link
• System of congruences, modular arithmetic. Link

Inconsummate Numbers

A number \(n\) is inconsummate if there is no number \(k\) which divided by its sum of digits gives \(n\). An example of a number that is not inconsummate would be 47 because there exists the number 846 such that 846 / (8+4+6) = 47. Notice that in order to confirm this, it was necessary to test numbers far ahead of 47 on the number line. The following SAGE code is designed to identify inconsummate numbers:

INPUT
# Generate a list of inconsummate numbers
# n is an inconsummate if no number k divided by its sum of digits gives n
# The larger the value of limit, the more accurate the list
limit=1000
inconsummate=[]
for number in range(1, limit):
    digits=(str(number))
    sum=0
    for n in range(0,len(digits)):
        sum+=Integer(digits[n])
    if number % sum == 0:
        inconsummate.append(number/sum)
X=Set(inconsummate)
Y=Set(range(1, limit+1))
print(Y.difference(X)) 

OUTPUT
{62, 63, 65, 66, 71, 72, 74, 75, ... }

Unfortunately after 65, the next inconsummate number is 75 and so 66, 71, 72 and 74 should be not included in the output. Notice that the limit was set to 1000. If we increase the limit to 10000, the output goes wrong here { ..., 432, 437, 438, ... } because 437 is not an inconsummate number. By setting the limit to 12000, this particular error is corrected. It would seem that by choosing a particular range (from 1 to the chosen limit), the resultant output is only reliable for about 1/30th of that range.

It's interesting to look at the frequency of inconsummate numbers:

up to \(10^1\) --> 0 --> 0%
up to \(10^2\) --> 6 --> 6%
up to \(10^3\) --> 111 --> 11.1%
up to \(10^4\) --> 1437 --> 14.37%
up to \(10^5\) --> 16430 --> 16.43%
up to \(10^6\) --> 183089 --> 18.3089%
up to \(10^7\) --> 1905285 --> 19.05285% seems to peak here
up to \(10^8\) --> 18907944 --> 18.907944%
up to \(10^9\) --> 183706706 --> 18.3706706%

Figure 1: 3 x 3 magic square 

For the ranges shown, the frequency reaches a peak of 19.05285% and then slowly decreases. The smallest 3 × 3 magic square whose entries are consecutive inconsummate numbers is shown in Figure 1. Source.

The initial inconsummate numbers are listed in OEIS A003635 and the initial members are:

62, 63, 65, 75, 84, 95, 161, 173, 195, 216, 261, 266, 272, 276, 326, 371, 372, 377, 381, 383, 386, 387, 395, 411, 416, 422, 426, 431, 432, 438, 441, 443, 461, 466, 471, 476, 482, 483, 486, 488, 491, 492, 493, 494, 497, 498, 516, 521, 522, 527, 531, 533, 536, ... 

Here is the full list of inconsummate numbers up to 40,000:

62, 63, 65, 75, 84, 95, 161, 173, 195, 216, 261, 266, 272, 276, 326, 371, 372, 377, 381, 383, 386, 387, 395, 411, 416, 422, 426, 431, 432, 438, 441, 443, 461, 466, 471, 476, 482, 483, 486, 488, 491, 492, 493, 494, 497, 498, 516, 521, 522, 527, 531, 533, 536, 542, 543, 546, 549, 552, 553, 563, 564, 573, 575, 612, 615, 620, 623, 630, 635, 639, 645, 650, 675, 716, 723, 750, 761, 771, 776, 785, 795, 816, 821, 822, 827, 831, 834, 839, 840, 845, 861, 866, 872, 876, 882, 926, 927, 932, 933, 936, 938, 941, 942, 945, 950, 966, 971, 972, 983, 986, 993, 1016, 1026, 1037, 1038, 1041, 1043, 1046, 1049, 1052, 1053, 1076, 1082, 1086, 1107, 1161, 1166, 1173, 1175, 1266, 1271, 1272, 1316, 1326, 1371, 1376, 1382, 1416, 1437, 1481, 1482, 1487, 1488, 1491, 1493, 1494, 1496, 1497, 1499, 1521, 1526, 1527, 1532, 1533, 1536, 1538, 1541, 1542, 1553, 1566, 1571, 1572, 1586, 1593, 1596, 1598, 1599, 1601, 1602, 1604, 1605, 1610, 1616, 1626, 1631, 1632, 1641, 1643, 1646, 1649, 1652, 1653, 1656, 1673, 1716, 1730, 1745, 1811, 1826, 1866, 1871, 1872, 1895, 1926, 1931, 1932, 1950, 1971, 1976, 1982, 1983, 1986, 2016, 2037, 2042, 2043, 2046, 2049, 2052, 2076, 2082, 2093, 2126, 2144, 2147, 2148, 2153, 2156, 2160, 2177, 2181, 2207, 2214, 2217, 2261, 2271, 2276, 2316, 2372, 2376, 2382, 2426, 2477, 2481, 2511, 2526, 2537, 2544, 2547, 2555, 2577, 2588, 2591, 2592, 2594, 2597, 2598, 2601, 2603, 2604, 2606, 2609, 2610, 2616, 2631, 2642, 2643, 2646, 2649, 2652, 2660, 2672, 2676, 2682, 2705, 2706, 2709, 2712, 2715, 2720, 2726, 2742, 2753, 2756, 2760, 2811, 2816, 2826, 2877, 2888, 2911, 2921, 2922, 2937, 2961, 2966, 2972, 2976, 2982, 2993, 3026, 3037, 3038, 3041, 3042, 3046, 3053, 3083, 3086, 3093, 3107, 3111, 3116, 3126, 3137, 3144, 3147, 3149, 3152, 3153, 3156, 3177, 3186, 3207, 3218, 3221, 3222, 3237, 3247, 3248, 3251, 3254, 3255, 3257, 3258, 3260, 3263, 3281, 3287, 3288, 3291, 3294, 3296, 3307, 3311, 3314, 3317, 3318, 3321, 3324, 3327, 3329, 3332, 3371, 3372, 3383, 3386, 3416, 3426, 3437, 3444, 3477, 3482, 3483, 3486, 3537, 3548, 3551, 3554, 3555, 3581, 3587, 3588, 3591, 3593, 3621, 3627, 3644, 3647, 3648, 3651, 3654, 3657, 3658, 3659, 3661, 3662, 3665, 3666, 3677, 3681, 3687, 3692, 3696, 3697, 3698, 3699, 3701, 3702, 3703, 3704, 3705, 3706, 3707, 3708, 3710, 3711, 3712, 3713, 3714, 3716, 3717, 3719, 3720, 3722, 3726, 3731, 3732, 3733, 3738, 3741, 3743, 3744, 3746, 3747, 3749, 3752, 3753, 3755, 3756, 3758, 3759, 3761, 3762, 3764, 3765, 3767, 3768, 3770, 3771, 3773, 3776, 3782, 3783, 3786, 3787, 3788, 3791, 3792, 3793, 3794, 3796, 3797, 3798, 3801, 3803, 3804, 3805, 3806, 3809, 3810, 3812, 3813, 3815, 3816, 3818, 3819, 3821, 3822, 3824, 3825, 3827, 3828, 3830, 3831, 3836, 3838, 3842, 3843, 3846, 3849, 3852, 3854, 3855, 3857, 3860, 3863, 3864, 3866, 3867, 3869, 3870, 3872, 3873, 3875, 3876, 3878, 3879, 3882, 3926, 3927, 3932, 3933, 3936, 3938, 3941, 3942, 3945, 3950, 3966, 3971, 3972, 3983, 3986, 3993, 4016, 4026, 4037, 4038, 4041, 4043, 4046, 4049, 4052, 4053, 4076, 4077, 4082, 4086, 4088, 4091, 4092, 4094, 4097, 4098, 4099, 4101, 4103, 4104, 4106, 4109, 4110, 4116, 4121, 4122, 4127, 4131, 4142, 4143, 4146, 4149, 4151, 4152, 4154, 4155, 4157, 4160, 4164, 4165, 4172, 4176, 4182, 4183, 4187, 4188, 4191, 4193, 4194, 4196, 4197, 4198, 4199, 4202, 4203, 4205, 4206, 4208, 4209, 4212, 4215, 4219, 4220, 4226, 4227, 4232, 4233, 4236, 4238, 4241, 4242, 4246, 4253, 4256, 4260, 4262, 4263, 4264, 4265, 4266, 4268, 4269, 4271, 4272, 4273, 4275, 4283, 4286, 4293, 4298, 4299, 4301, 4302, 4304, 4305, 4308, 4309, 4310, 4313, 4316, 4319, 4320, 4323, 4326, 4338, 4341, 4343, 4346, 4349, 4352, 4353, 4356, 4371, 4373, 4376, 4378, 4379, 4380, 4382, 4407, 4409, 4410, 4412, 4415, 4419, 4430, 4437, 4461, 4466, 4471, 4472, 4476, 4482, 4487, 4488, 4491, 4493, 4494, 4496, 4497, 4499, 4521, 4526, 4527, 4532, 4533, 4536, 4538, 4541, 4542, 4553, 4566, 4571, 4572, 4586, 4593, 4594, 4596, 4597, 4598, 4599, 4601, 4602, 4604, 4605, 4610, 4616, 4626, 4631, 4632, 4641, 4642, 4643, 4646, 4649, 4652, 4653, 4656, 4660, 4671, 4676, 4682, 4704, 4705, 4706, 4709, 4710, 4712, 4715, 4737, 4742, 4743, 4746, 4749, 4752, 4760, 4763, 4782, 4793, 4811, 4814, 4815, 4817, 4818, 4820, 4821, 4823, 4824, 4826, 4827, 4829, 4830, 4832, 4836, 4838, 4841, 4842, 4853, 4854, 4856, 4857, 4859, 4860, 4862, 4863, 4865, 4866, 4868, 4869, 4871, 4872, 4875, 4878, 4880, 4892, 4893, 4896, 4898, 4899, 4901, 4902, 4904, 4905, 4910, 4912, 4919, 4920, 4923, 4926, 4928, 4929, 4930, 4931, 4932, 4934, 4935, 4940, 4946, 4949, 4952, 4953, 4956, 4964, 4965, 4967, 4970, 4973, 4976, 4979, 4980, 4982, 4989, 5016, 5037, 5038, 5042, 5043, 5046, 5049, 5052, 5076, 5082, 5093, 5126, 5144, 5147, 5148, 5153, 5156, 5160, 5177, 5181, 5187, 5192, 5196, 5198, 5199, 5201, 5202, 5204, 5205, 5210, 5220, 5226, 5231, 5232, 5252, 5253, 5256, 5262, 5264, 5265, 5267, 5270, 5282, 5283, 5286, 5297, 5298, 5301, 5303, 5304, 5305, 5306, 5309, 5310, 5312, 5313, 5315, 5316, 5319, 5330, 5336, 5342, 5343, 5346, 5349, 5352, 5360, 5363, 5372, 5373, 5375, 5376, 5378, 5379, 5382, 5393, 5409, 5412, 5415, 5420, 5423, 5426, 5430, 5453, 5456, 5460, 5474, 5477, 5481, 5489, 5490, 5507, 5514, 5517, 5520, 5530, 5537, 5562, 5564, 5565, 5573, 5595, 5612, 5613, 5615, 5619, 5630, 5634, 5639, 5640, 5645, 5673, 5675, 5723, 5730, 5745, 5750, 5777, 5811, 5816, 5826, 5877, 5881, 5888, 5921, 5922, 5927, 5931, 5933, 5936, 5942, 5943, 5945, 5966, 5972, 5976, 5982, 5993, 6026, 6037, 6038, 6041, 6042, 6053, 6086, 6093, 6107, 6113, 6119, 6120, 6123, 6150, 6174, 6185, 6189, 6200, 6221, 6222, 6230, 6284, 6285, 6300, 6326, 6335, 6339, 6345, 6350, 6371, 6372, 6376, 6389, 6390, 6416, 6426, 6437, 6444, 6450, 6477, 6482, 6483, 6486, 6500, 6537, 6548, 6551, 6554, 6555, 6581, 6587, 6588, 6591, 6593, 6621, 6627, 6637, 6644, 6647, 6648, 6651, 6654, 6657, 6659, 6662, 6665, 6675, 6716, 6723, 6750, 6777, 6816, 6881, 6887, 6921, 6926, 6977, 6981, 6987, 6992, 6996, 6997, 6998, 6999, 7011, 7022, 7031, 7032, 7033, 7037, 7038, 7041, 7043, 7046, 7049, 7052, 7053, 7076, 7082, 7086, 7107, 7116, 7126, 7137, 7148, 7149, 7152, 7160, 7181, 7214, 7217, 7223, 7230, 7277, 7284, 7295, 7311, 7322, 7331, 7332, 7377, 7395, 7421, 7422, 7427, 7431, 7442, 7445, 7476, 7482, 7487, 7500, 7521, 7526, 7527, 7532, 7533, 7536, 7538, 7541, 7542, 7553, 7566, 7571, 7572, 7586, 7593, 7596, 7598, 7599, 7601, 7602, 7604, 7605, 7610, 7616, 7626, 7631, 7632, 7641, 7643, 7646, 7649, 7652, 7653, 7656, 7671, 7676, 7682, 7704, 7706, 7709, 7710, 7712, 7715, 7737, 7742, 7752, 7760, 7784, 7785, 7821, 7826, 7827, 7832, 7835, 7839, 7845, 7850, 7866, 7871, 7872, 7895, 7926, 7931, 7932, 7950, 7976, 7982, 8037, 8042, 8076, 8082, 8093, 8126, 8144, 8147, 8148, 8151, 8153, 8154, 8156, 8157, 8159, 8160, 8162, 8165, 8171, 8172, 8187, 8192, 8196, 8198, 8199, 8201, 8202, 8204, 8205, 8210, 8220, 8226, 8231, 8232, 8252, 8253, 8256, 8262, 8264, 8265, 8267, 8270, 8282, 8306, 8309, 8310, 8312, 8315, 8334, 8339, 8340, 8345, 8372, 8376, 8382, 8390, 8400, 8426, 8444, 8445, 8450, 8477, 8481, 8483, 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64733, 64736, 64737, 64744, 64748, 64751, 64754, 64755, 64757, 64758, 64761, 64764, 64766, 64767, 64769, 64770, 64772, 64773, 64775, 64776, 64781, 64787, 64788, 64791, 64794, 64796, 64797, 64799, 64802, 64803, 64804, 64805, 64806, 64807, 64808, 64809, 64811, 64812, 64814, 64815, 64820, 64823, 64826, 64830, 64844, 64847, 64848, 64851, 64853, 64856, 64860, 64863, 64874, 64877, 64881, 64884, 64887, 64893, 64907, 64914, 64917, 64922, 64925, 64926, 64937, 64947, 64955, 64959, 64962, 64974, 64977, 64985, 64992, 64995, 65000, 65021, 65022, 65027, 65031, 65033, 65036, 65037, 65048, 65051, 65054, 65055, 65061, 65066, 65072, 65077, 65081, 65087, 65088, 65091, 65094, 65096, 65097, 65099, 65102, 65103, 65104, 65105, 65106, 65109, 65121, 65127, 65132, 65137, 65141, 65142, 65144, 65147, 65148, 65151, 65154, 65157, 65159, 65162, 65165, 65171, 65172, 65177, 65181, 65207, 65214, 65217, 65226, 65237, 65255, 65259, 65288, 65291, 65292, 65294, 65307, 65318, 65321, 65322, 65324, 65325, 65327, 65328, 65337, 65347, 65348, 65351, 65354, 65355, 65357, 65364, 65366, 65367, 65369, 65370, 65381, 65384, 65385, 65387, 65391, 65394, 65396, 65397, 65399, 65402, 65403, 65405, 65406, 65407, 65414, 65417, 65418, 65421, 65424, 65427, 65429, 65432, 65435, 65437, 65439, 65442, 65447, 65448, 65451, 65454, 65457, 65459, 65462, 65465, 65474, 65475, 65477, 65478, 65480, 65481, 65484, 65487, 65492, 65495, 65501, 65502, 65504, 65505, 65507, 65510, 65514, 65517, 65525, 65528, 65529, 65532, 65534, 65535

on 29th August 2021 and again on 19th February 2023