Do you have Forms Professional and Lookups? If so, I'd probably do something like this:

1. Store the numbers in a SQL database table
2. Connect that table to the form as a Data Source
3. Create a field on the form called "Validation" or something like that and give it a custom CSS class of "inputValidation" or whatever makes sense to you.
4. Make the field Read Only and use Field Rules to hide it always
5. Create a lookup for the validation field (when Input = Column, populate Validation Field with Column)
6. Assign a custom validator to your input field that require the lookup to be blank.

$(document).ready(function(){
  // assign custom validator
  customValidators();
  
  // trigger validation of input field when validation field changes
  $('.inputValidation input').on('change',function(){
    $('#Field37').parsley().validate();
  });
});

function customValidators(){
  // validator to check if value is not found by lookup
  window.Parsley.addValidator('eligiblevalue', {
    validateString: function(value) {
      return value != $('.inputValidation input').val();
    },
    messages: {
      en: 'It appears you are not eligible.'
    }
  });
  
  // assign validator to source field
  $('#Field37').attr('data-parsley-eligiblevalue','');
}

Now, when the user enters something that matches an entry in the table, it will populate the validation field, trigger the validation on the input field, and display an error directly under the field. The added bonus is that the validation error will prevent submission without having to hide the button.

Note that with the pop-up message and showing/hiding the button, there's a still a chance that something could still slip through and get submitted when it shouldn't have.

A database is probably the best option in case you need to change the values later, but you could use a similar approach even without that if you don't have lookups.