You can look at the sum of the digits and if that sum of the digits is divisible by 3 then the original number is divisible by 3.And so if it’s divisible by 3 in base 10 it’s also divisible by 3 in base 4.
My friend Patrick Paterson had a patented method for generating bigger primes. He asked me to give him my favorite prime number, so I gave him 17. He then wrote the number in base 4 and noticed that if you read it as if it’s base 10, 101 is also a prime in base 10. He tried it with other prime numbers and noticed that no matter what prime number he plugged in, he always seemed to get out another prime number. For example, when he tried it with 7, he got 13; when he tried it with 11, he got another prime. He then tried it with 101, and got 1211 which is not a prime, but 7 times 173. He noticed that for the small numbers, the rule seemed to work, but it broke down for larger ones. He realized that if the number is even in base 10, it’s also even in base 4, and if it’s divisible by 3 in base 10, it’s also divisible by 3 in base 4. If someone asks you if a number is divisible by 3, you can use the Paterson technique to answer. This technique involves adding up all of the digits of the number, and if the sum is divisible by 3, then the original number is also divisible by 3. This works because the base that we are using, 10, is one above a multiple of 3. Additionally, operations like multiplication and addition play nicely with reduction when reducing mod 3. This means that if you have two very large numbers, you can take the remainder of each before adding them to find the remainder of their sum. The same applies for products. When expanding a large number, 10 is just 1 mod 3, so you can think of the expansion as 9 times 1 to the power 3, plus 7 times 1 to the power 2, and 3 times 1 to the power 1 and 2. This means that you can add the digits to find the reduction of their sum. This same rule applies if you write the number in base 4. This means that numbers are immune to divisibility by 2, 3 and 5, but not by 7, 13 or 11. To sum up, this video discusses the Strong Law of Small Numbers which states that small numbers cannot support the weight of all the patterns they have to hold and coincidences can occur. Grant’s channel, 3Blue1Brown, has released a video about false patterns and shows the first 1000 prime numbers in base 10 converted to base 4 and checked for Paterson primality. Out of these 1000 prime numbers, only 388 passed Patrick’s test. More information can be found on the links provided in the video description.