nt.number theory – Finite groups arising as Galois groups of maximal unramified extension of number fields

I was wondering if it is known for which number fields the maximal unramified (non-abelian) extension is of finite degree or do we know the finite groups that arise as the Galois groups of these finite degree maximal unramified extensions.

I have seen the trivial group and the group of two elements as these Galois groups but not beyond that.

Thanks in advance.

PG query to calculate how many special number is in a column

I have a table like this:

id —value






I need to count all rows in this order:
each rows with id of n should be count except if previous value of that row are repeated in previous rows, for example in above sample rows one, two and four should be count but rows tree and five should not be count (value in row three and five are equal to row two) and the result should be 3, is there any way to do this with PG SQL?

research – Relationship between the prime counting function of some number and the sum of prime numbers less than the square root of that number

A little bit more than a year ago I was able to prove that the sum of the prime numbers less than the square root of some number was asymptotically equivalent to the number of primes up that number (see http://vixra.org/abs/1911.0316).

In this same pre-print, I studied the numbers for which the number of prime numbers up to some number was exactly equal to the sum of the prime numbers less than the square root of that number. As a result, I derived the set of prime numbers that you can see here (http://oeis.org/A329403) and conjectured that there existed infinitely many of such prime numbers. This would imply that the value of the prime counting function of a given number would be infinitely often equal to the sum of prime numbers up to the square root of that number, and in other cases the sum described would be a very good estimator of the prime counting function.

A little bit more than a year after, I am unable to prove or disprove this conjecture. Unfortunately, as an amateur mathematician I feel uncapable of bringing this interesting research further, but it would be great if someone were able to prove or disprove the conjecture.

I find the relationship exposed between sums of prime numbers and the prime counting function really appealling and misterious, and I feel that there might be some hidden and profound mathematics there that I am not able to grasp.

I would like to hear your thoughts and ideas (if any) of how to bring the research to a happy ending. Thanks in advance for your time and effort!

php – Need help with generating a unique reference number based on the user id

I’m trying to create a unique reference ID based on the actual user ID so I don’t have to reveal the actual user ID on my site. This number will be published later using a shortcode. It is necessary to hide the real user ID and real name due to the privacy policy of my country. I was thinking of a user meta that is automatically generated after registration based on the actual user ID with a specific pattern:

  • 8-digits
  • contains the actual user ID
  • automatically completes the remaining digits with randomly generated numbers for fixing number length
{user_id}{generated_nr}, max 8-digits

I started very simply as follows, but I can’t set the length/width of the number considering the user id, so the missing digits are filled with auto-generated numbers. I also do not know how to generate the number only for newly registered users, nor do I know how to check if the number has already been assigned to another user:

function add_custom_ref_id( $user_id ) {

 $custom_ref_id = $user_id . wp_rand(10000000,99999999);
 return $custom_ref_id;


add_user_meta( $user_id, 'custom_ref_id', $custom_ref_id );

Concerns: Will the numbers be unique? Is this method secure? How can I avoid mistakes? Maybe combining them with the first letters of their first and last name? Like;

Name: John Doe, User ID: 24 > 24XXXXXXJD

And if yes, how?

I am always open to new suggestions and thank you in advance for proposing solutions.

Best regards.

PS: I already read the older threads but apart from the fact that they do not really meet my expectations, there could be better, more secure and more efficient methods out there.

Display line number where error occurred in DB2 stored procedure

I have added an EXIT handler in my procedure that captures the SQLSTATE and SQLCODE, and even found a way to get the procedure name, but I also need to know where the error occurred. Suggestions greatly appreciated.

declare EXIT handler for SQLEXCEPTION
    select sysibm.routine_specific_name, SQLSTATE, SQLCODE 
    into v_sp_name, v_sqlstate, v_sqlcode 
    from sysibm.sysdummy1;

    call dbms_output.put_line('Error in '||v_sp_name ' ('||v_sqlstate, v_sqlcode||')');

algorithms – What is the Number of epochs with no improvement after which training will be stopped.?

I am trying to make a Convolutional neural network. Training the images of different brands of Logos. Have 100 images per class and there are 40 classes. I have trained the model now want to check that model is overfitted or not . What should be the number of epochs we will se that with no improvement after which training will be stopped . Should I see after 2, 3 or which number ?

python – Determine whether a number and its square are both the sum of two squares

You can speed it up by saving the squares in a set. If you wrote a function to do this, you can give it a mutable default argument for it to save all the squares instead of having to calculate them repeatedly.

# dict used because there’s no insertion ordered set
def sum_squares(n, saved_squares=dict()):
    # check for sum while saving squares for future calls
    check_end = n
    if int(n**(1/2)) > len(saved_squares):
        check_end = (len(saved_squares) + 1)**2
        found_while_saving = False
        for num_to_save in range(len(saved_squares) + 1, int(n**(1/2) + 1)):
            square = num_to_save**2
            saved_squares(square) = None
            if n - square in saved_squares:
                found_while_saving = True
        if found_while_saving == True:
            return True
    # check for sum in set of already calculated squares
    for square in saved_squares:
        if n - square in saved_squares:
            return True
        # insertion order needed so early break won’t happen
        if square >= check_end:
    return False

With the numbers 5881, 2048, 2670, and 3482, it will only have to calculate any squares for the first call with the number 5881. On the other three calls it can skip to checking the set of all the already calculated squares. But no matter what order of numbers it’s called with, it will only have to calculate the square of any number once.

This is how I tested it on repl.it:

nums_to_check = (2048, 2670, 3482, 5881)
for n in nums_to_check:
    print(sum_squares(n) and sum_squares(n**2))

It completed in under a second, so hopefully it’s fast enough for the site you’re using.