15697673 37784951 37679651 15948773
11012041 32223013 14021011 31032223
10203031 33054023 13030201 32045033

Faride FiroozBakht found other interesting examples:

14^2+87^2=41^2+78^2
104^2+807^2=401^2+708^2
1004^2+8007^2=4001^2+7008^2
10004^2+80007^2=40001^2+70008^2
...

27^2+96^2=72^2+69^2
207^2+906^2=702^2+609^2
2007^2+9006^2=7002^2+6009^2
20007^2+90006^2=7000^2+60009^2
...

4^2+53^2=40^2+35^2
4^2+503^2=400^2+305^2
4^2+5003^2=4000^2+3005^2
4^2+50003^2=40000^2+30005^2
...

3^2+54^2=30^2+45^2
3^2+504^2=300^2+405^2
3^2+5004^2=3000^2+4005^2
3^2+50004^2=30000^2+40005^2

***

More solutions for Q.2 came from Jon Wharf (5/1/2003)

question 2 asks for readily extensible pairs (plus reverses) with a potential to produce prime numbers.

One example is:

303949^2 + 959333^2 = 949303^2 + 333959^2
3030949^2 + 9590333^2 = 9490303^2 + 3330959^2
30300949^2 + 95900333^2 = 94900303^2 + 33300959^2
303000949^2 + 959000333^2 = 949000303^2 + 333000959^2

How did I look for them?

For such a LH pair of numbers with 2n digits a & b, break down into n digit numbers as follows: a=w*10^n+x and b=y*10^n+z

Now w,x,y,z must fulfill the following [rev() is the digit reverse]:

1. for the coprime-to-30 condition:

a) rev(w), x, rev(y), z all <> 0 mod 2 and <> 0 mod 5;
b) w+x and y+z <>0 mod 3

2. for the zero-inflatable reverse-sum-square equal, we want (w*k+x)^2 + (y*k+z)^2 = (rev(x)*k+rev(w))^2 + (rev(z)*k+rev(y))^2 k=10^m, m>=n, and collecting like powers of k we get

a) w^2+y^2 = rev(x)^2+rev(z)^2
b) w*x+y*z = rev(x)*rev(w)+rev(z)*rev(y)
c) x^2+z^2 = rev(w)^2+rev(y)^2

Without loss of generality, I also set w < rev(x) < y . This ensures rev(z) > w also.

So I trawled through the combinations of w, x, y and saw if the resulting z was any good - geting two extensible pairs for six digits and another 37 for eight digits.

Six digit pairs and reverses:

111373, 383141, 373111, 141383
303949, 959333, 949303, 333959 *

Eight digit pairs and reverses:

10003231, 37654663, 13230001, 36645673 *
10122343, 36722561, 34322101, 16522763
10162303, 32926361, 30326101, 16362923 *
10203031, 33054023, 13030201, 32045033 *#
10221233, 35612461, 33212201, 16421653
10431043, 37913791, 34013401, 19731973
11012221, 14421431, 12221011, 13412441
11133353, 37933771, 35333111, 17733973
11240233, 35935771, 33204211, 17753953 *
11331133, 35713571, 33113311, 17531753
11430023, 34603471, 32003411, 17430643
12002641, 15620231, 14620021, 13202651
12023231, 15632641, 13232021, 14623651
12140323, 33735551, 32304121, 15553733 *
12303551, 37754563, 15530321, 36545773
12342143, 36924781, 34124321, 18742963
12584723, 33917351, 32748521, 15371933 *
12853463, 38904781, 36435821, 18740983 *
13503661, 36654353, 16630531, 35345663 *
13603561, 34354033, 16530631, 33045343 *
13662653, 36926761, 35626631, 16762963 *
13753553, 36704561, 35535731, 16540763
14471843, 34948741, 34817441, 14784943 *
14814971, 39787483, 17941841, 38478793 *
14831483, 39713971, 38413841, 17931793
15640673, 37735951, 37604651, 15953773
15803981, 36754253, 18930851, 35245763 *
30039449, 95593333, 94493003, 33339559
30102523, 38520563, 32520103, 36502583
30201413, 37410463, 31410203, 36401473
30940349, 95335933, 94304903, 33953359 *
31113533, 39731773, 33531113, 37713793
31206673, 38660423, 37660213, 32406683 *
31414243, 92487419, 34241413, 91478429 *
32136253, 37463543, 35263123, 34536473 *
32223433, 39832883, 33432223, 38823893
39040439, 93535393, 93404093, 39353539 *

Note that one of the eight digit sets (the fourth, #) is Jud McCranie's third solution.

Looking at the definition of k, some of the pairs lend themselves to overlapping the upper and lower half (ie. k=10^m, m<n), which is OK provided the digits concerned don't sum to more than 10. Unfortunately it doesn't produce any new complete prime sets.

Also I realised that increasing m by 2 (two extra zeros) doesn't change a number mod 11. So we can eliminate almost half the pairs by checking divisibility by 11 for say m=4 and 5. This reduces the 8-digit pairs to check to only 18 pairs, starred above, and only one six digit pair remains.

I also removed {10431043,37913791} - although it could still work mod 11, it will never produce primes.

So, having done that, I searched through to m=1004 - 1000 inserted zeros - using pfgw. Disappointly, there were no new quartets of primes, and only two near-misses:

3866*10^16+0423
3766*10^16+0213
3240*10^16+6683

and

3213*10^17+6253
3526*10^17+3123
3453*10^17+6473

So, although the question is answered, no more prime sets were found.

113123333234444432333324141
361667359458888854953765153
141423333234444432333321311
351567359458888854953766163

124231133333444333331135251
392791195379848973591196283
152531133333444333331132421
382691195379848973591197293

135014425851000158524413661
366534467873000378764434353
166314425851000158524410531
353434467873000378764435663