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Pyth, 1415 bytes

}Qmsjk^R2d*SQUQmjk^R2d*SsQUs

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Test casesTest cases

A program that takes an integer value as a string (because the trailing 0 is recognized as a a separate value) and outputs True for the 'concats of two squares', False otherwise. The calculation time for the large values exceeds 1 minute.

Explanation:

}Q              # check if the input value is in the list.
  m             # The list is constructed by mapping:
   s            # numeric conversion of the strings,
    jk           # concatenation
      ^R2        # and squaring
         d       # of the argument d
          *      # mapped over the cartesian product
           SQSsQ   # of the range from 1 to Q
             U Us # with the range from 0 to Q-1

Pyth, 14 bytes

}Qmsjk^R2d*SQU

Try it online!

Test cases

A program that takes an integer value and outputs True for the 'concats of two squares', False otherwise. The calculation time for the large values exceeds 1 minute.

Explanation:

}Q              # check if the input value is in the list.
  m             # The list is constructed by mapping:
   s            # numeric conversion of the strings,
    jk          # concatenation
      ^R2       # and squaring
         d      # of the argument d
          *     # mapped over the cartesian product
           SQ   # of the range from 1 to Q
             U  # with the range from 0 to Q-1

Pyth, 15 bytes

}Qmjk^R2d*SsQUs

Try it online!

Test cases

A program that takes an integer value as a string (because the trailing 0 is recognized as a a separate value) and outputs True for the 'concats of two squares', False otherwise. The calculation time for the large values exceeds 1 minute.

Explanation:

}Q              # check if the input value is in the list.
  m             # The list is constructed by mapping:    
   jk           # concatenation
     ^R2        # and squaring
        d       # of the argument d
         *      # mapped over the cartesian product
          SsQ   # of the range from 1 to Q
             Us # with the range from 0 to Q-1
deleted 3 characters in body
Source Link

Pyth, 1514 bytes

}Qmsjk^R2d*SQUhQmsjk^R2d*SQU

Try it online!Try it online!

Test casesTest cases

A program that takes an integer value and outputs True for the 'concats of two squares', False otherwise. The calculation time for the large values exceeds 1 minute.

Explanation:

}Q              # check if the input value is in the list.
  m             # The list is constructed by mapping:
   s            # numeric conversion of the strings,
    jk          # concatenation
      ^R2       # and squaring
         d      # of the argument d
          *     # mapped over the cartesian product
           SQ   # of the range from 1 to Q
             UhU  # with the range from 0 to Q-1

Pyth, 15 bytes

}Qmsjk^R2d*SQUh

Try it online!

Test cases

A program that takes an integer value and outputs True for the 'concats of two squares', False otherwise. The calculation time for the large values exceeds 1 minute.

Explanation:

}Q              # check if the input value is in the list.
  m             # The list is constructed by mapping:
   s            # numeric conversion of the strings,
    jk          # concatenation
      ^R2       # and squaring
         d      # of the argument d
          *     # mapped over the cartesian product
           SQ   # of the range from 1 to Q
             Uh # with the range from 0 to Q

Pyth, 14 bytes

}Qmsjk^R2d*SQU

Try it online!

Test cases

A program that takes an integer value and outputs True for the 'concats of two squares', False otherwise. The calculation time for the large values exceeds 1 minute.

Explanation:

}Q              # check if the input value is in the list.
  m             # The list is constructed by mapping:
   s            # numeric conversion of the strings,
    jk          # concatenation
      ^R2       # and squaring
         d      # of the argument d
          *     # mapped over the cartesian product
           SQ   # of the range from 1 to Q
             U  # with the range from 0 to Q-1
Source Link

Pyth, 15 bytes

}Qmsjk^R2d*SQUh

Try it online!

Test cases

A program that takes an integer value and outputs True for the 'concats of two squares', False otherwise. The calculation time for the large values exceeds 1 minute.

Explanation:

}Q              # check if the input value is in the list.
  m             # The list is constructed by mapping:
   s            # numeric conversion of the strings,
    jk          # concatenation
      ^R2       # and squaring
         d      # of the argument d
          *     # mapped over the cartesian product
           SQ   # of the range from 1 to Q
             Uh # with the range from 0 to Q