There are 8363 5-digit primes and 217 5-digit squares.

Loop through every possible pair, note the sum and make a dictionary where the keys are the various sums, and each value is a list of the lists of [prime,square] pairs which add to that sum.

Once the dictionary is formed, find which key (which sum) corresponds to the most pairs.  This will be the upper limit on the number of lecture attendees.

It turns out that 101888 is the sum that corresponds to the most (prime,square) pairs; there are 69 of them.

There are no sums that share more than 69 such pairs.

There were 69 attendees already, Lenny would have made 70 and it would have been impossible.  The attendees' cards all summed to  101888

101888 [[10079, 91809], [11287, 90601], [12487, 89401], [13679, 88209], [17207, 84681], [18367, 83521], [20663, 81225], [21799, 80089], [26263, 75625], [28447, 73441], [29527, 72361], [31663, 70225], [32719, 69169], [33767, 68121], [34807, 67081], [35839, 66049], [37879, 64009], [39887, 62001], [40879, 61009], [41863, 60025], [42839, 59049], [46663, 55225], [47599, 54289], [48527, 53361], [50359, 51529], [51263, 50625], [53047, 48841], [53927, 47961], [54799, 47089], [55663, 46225], [56519, 45369], [57367, 44521], [58207, 43681], [59863, 42025], [60679, 41209], [61487, 40401], [63079, 38809], [63863, 38025], [65407, 36481], [66919, 34969], [68399, 33489], [69127, 32761], [69847, 32041], [71263, 30625], [72647, 29241], [73327, 28561], [73999, 27889], [75967, 25921], [76607, 25281], [77239, 24649], [77863, 24025], [78479, 23409], [79087, 22801], [79687, 22201], [80279, 21609], [80863, 21025], [81439, 20449], [82007, 19881], [82567, 19321], [83663, 18225], [84199, 17689], [85247, 16641], [86263, 15625], [88663, 13225], [89119, 12769], [89567, 12321], [90007, 11881], [90439, 11449], [90863, 11025]]

----------------

def isprime(n):

    '''check if integer n is a prime'''

    n = abs(int(n))

    if n < 2:

        return False

    if n == 2: 

        return True    

    if not n & 1: 

        return False

    for x in range(3, int(n**0.5)+1, 2):

        if n % x == 0:

            return False

    return True

primes = [p for p in range(10001,99999,2) if isprime(p)]

squares = [i**2 for i in range(100,317)]

sumDict = {}

for p in primes:

    for s in squares:

        x = p+s

        if x not in sumDict:

            sumDict[x] = [[p,s]]

        else:

            sumDict[x].append([p,s])

lengths = []

for a in sumDict:

    lengths.append(len(sumDict[a]))

longest = max(lengths)  

print(longest) 

    

for a in sumDict:   

    if len(sumDict[a]) == longest:

        print(a, sumDict[a])