# The code below focuses on record values for x in x^2-py^2=2.
# PLy is the known initial segment of the record list
# startp is the prime to start the search at
# m is the number of rounds to go, cycling through larger primes.
def recordx2primelistextend(PL, startp, m):
    pr = startp                                     # Set the beginning prime value
    k = len(PL)                                     # Compute the number of items in the given list
    print("Number of initial data points is", k)    # Announce the current number of data points
    R = PL[k-1][1]                                  # Find the current record x-value in the list PL
    p = PL[k-1][0]                                  # Find the current record prime in the list PL
    NEW = 0                                         # Set the number of new data points at 0
    NList = []                                      # Initiate an empty list for the new data
    for i in range(m+1):                            # Start running through the next m prime numbers
        pr = next_prime(pr)                         # Increase the prime number from the current value
        K.<sqrtpr> = QuadraticField(pr)             # Call on the mathematical technology needed
        cf = continued_fraction(sqrtpr);            # Find the continued fraction of sqrt of prime
        kk = cf.period()                            # Find the period of the continued fraction
        mm = len(kk)+1                              # Find the length mm of the period
        for i in range(2*mm+1):                     # Search solution among first 2mm+1 convergents
            ppr = cf.convergent(i).numerator()      # x-value of convergent i
            qqr = cf.convergent(i).denominator()    # y-value of the convergent i
            if (ppr^2 - pr*qqr^2 == 2):             # If x^2 - p y^2 = 2, x is a candidate
                if (ppr>R):                         # If this x exceeds the current record value,
                    NList = NList + [(pr, ppr)]      # extend the new data list by a new term (p, x)
                    R = ppr                         # Update the record value of x
                    NEW = NEW+1                     # Increase the new data count by 1
                    break
                else:                               # If the x is not a record,
                    break                           # End the search for this prime number
    print("Number of new data points", NEW)          # When all is done, say how many new data points
    print("Start Prime", pr)
    return(pr, NList)