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Numbers which are simultaneously of the form x^2+y^2, x^2+2y^2, x^2+3y^2, x^2+7y^2, all with x>=0, y>=0.
+10
9
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 193, 196, 225, 256, 289, 324, 337, 361, 400, 441, 457, 484, 529, 576, 625, 673, 676, 729, 772, 784, 841, 900, 961, 1009, 1024, 1033, 1089, 1129, 1156, 1201, 1225, 1296, 1297, 1348, 1369, 1444, 1521, 1600, 1681, 1737, 1764, 1801, 1828, 1849, 1873, 1936, 2017
Let S_k = {x^2+k*y^2: x,y positive integers}. How many out of S_1, S_2, S_3, S_7 does n belong to?
+10
7
0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 2, 0, 0, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 2, 1, 2, 0, 2, 3, 1, 1, 1, 2, 0, 3, 2, 1, 0, 0, 2, 1, 1, 1, 2, 2, 1, 0, 1, 2, 1, 1, 0, 2, 0, 1, 2, 1, 1, 3, 2, 0, 0, 1, 3, 3, 1, 1, 2, 1, 0, 2, 1, 1, 2, 1, 1, 1, 1, 0, 2, 2, 1, 1, 1, 1, 0, 0, 1, 3, 1, 2, 2, 1, 1, 1, 1, 0, 1, 2, 2, 3, 0, 1, 2, 3, 1, 0, 2, 2, 1, 0, 0
COMMENTS
"If a composite number C is of the form a^2 + kb^2 for some integers a & b, then every prime factor P (of C) raised to an odd power is of the form c^2 + kd^2, for some integers c & d."
This statement is only true for k = 1, 2, 3. For k = 7, with the exception of the prime factor 2, the statement mentioned above is true.
A number can be written as a^2 + b^2 if and only if it has no prime factor congruent to 3 (mod 4) raised to an odd power.
A number can be written as a^2 + 2b^2 if and only if it has no prime factor congruent to 5 (mod 8) or 7 (mod 8) raised to an odd power.
A number can be written as a^2 + 3b^2 if and only if it has no prime factor congruent to 2 (mod 3) raised to an odd power.
A number can be written as a^2 + 7b^2 if and only if it has no prime factor congruent to 3 (mod 7) or 5 (mod 7) or 6 (mod 7) raised to an odd power and the exponent of 2 is not 1.
PROG
(PARI) for(n=1, 100, sol=0; for(x=1, 100, if(issquare(n-x*x)&&n-x*x>0, sol++; break)); for(x=1, 100, if(issquare(n-2*x*x)&&n-2*x*x>0, sol++; break)); for(x=1, 100, if(issquare(n-3*x*x)&&n-3*x*x>0, sol++; break)); for(x=1, 100, if(issquare(n-7*x*x)&&n-7*x*x>0, sol++; break)); print1(sol", ")) /* V. Raman, Oct 16 2012 */
Let S_k = {x^2+k*y^2: x,y nonnegative integers}. How many out of S_1, S_2, S_3, S_7 does n belong to?
+10
7
4, 2, 2, 4, 1, 1, 2, 3, 4, 1, 2, 2, 2, 0, 0, 4, 2, 2, 2, 1, 1, 1, 1, 1, 4, 1, 2, 2, 2, 0, 1, 3, 1, 2, 0, 4, 3, 1, 1, 1, 2, 0, 3, 2, 1, 0, 0, 2, 4, 2, 1, 2, 2, 1, 0, 1, 2, 1, 1, 0, 2, 0, 2, 4, 1, 1, 3, 2, 0, 0, 1, 3, 3, 1, 2, 2, 1, 0, 2, 1, 4, 2, 1, 1, 1, 1, 0, 2, 2, 1, 1, 1, 1, 0, 0, 1, 3, 2, 2, 4, 1, 1, 1, 1, 0, 1, 2, 2, 3, 0, 1, 2, 3, 1, 0, 2, 2, 1, 0, 0
COMMENTS
"If a composite number C is of the form a^2 + kb^2 for some integers a & b, then every prime factor of C raised to an odd power is of the form c^2 + kd^2 for some integers c & d."
This statement is only true for k = 1, 2, 3.
For k = 7, with the exception of the prime factor 2, the statement mentioned above is true.
A number can be written as a^2 + b^2 if and only if it has no prime factor congruent to 3 (mod 4) raised to an odd power.
A number can be written as a^2 + 2b^2 if and only if it has no prime factor congruent to 5 (mod 8) or 7 (mod 8) raised to an odd power.
A number can be written as a^2 + 3b^2 if and only if it has no prime factor congruent to 2 (mod 3) raised to an odd power.
A number can be written as a^2 + 7b^2 if and only if it has no prime factor congruent to 3 (mod 7) or 5 (mod 7) or 6 (mod 7) raised to an odd power, and the exponent of 2 is not 1.
Comment from N. J. A. Sloane, Sep 14 2012: S_1, S_2, S_3, S_7 are the first four quadratic forms with class number 1. (See Cox, for example.)
REFERENCES
David A. Cox, Primes of the Form x^2 + n y^2, Wiley, 1989. - From N. J. A. Sloane, Sep 14 2012
PROG
(PARI) for(n=1, 100, sol=0; for(x=0, 100, if(issquare(n-x*x)&&n-x*x>=0, sol++; break)); for(x=0, 100, if(issquare(n-2*x*x)&&n-2*x*x>=0, sol++; break)); for(x=0, 100, if(issquare(n-3*x*x)&&n-3*x*x>=0, sol++; break)); for(x=0, 100, if(issquare(n-7*x*x)&&n-7*x*x>=0, sol++; break)); print1(sol", ")) /* V. Raman, Oct 16 2012 */
Numbers which can be written neither as a^2+b^2, nor as a^2+2*b^2, nor as a^2+3*b^2, nor as a^2+7*b^2, with a >= 0 and b >= 0.
+10
4
14, 15, 30, 35, 42, 46, 47, 55, 60, 62, 69, 70, 78, 87, 94, 95, 105, 110, 115, 119, 120, 126, 135, 138, 140, 141, 142, 143, 154, 155, 158, 159, 165, 167, 168, 174, 182, 186, 188, 190, 195, 206, 210, 213, 215, 220, 222, 230, 231, 235, 238, 240, 248, 254, 255, 266, 270, 276, 280, 282, 285, 286, 287, 295, 299
COMMENTS
If a composite number C, in case, can be written in the form C = a^2+k*b^2, for some integers a & b, then every prime factor P (for C) being raised to an odd power can be written in the form P = c^2+k*d^2, for some integers c & d.
This statement is only true for k = 1, 2, 3.
For k = 7, with the exception of the prime factor 2, the statement mentioned above is true.
Numbers which can be written neither as a^2+b^2, nor as a^2+2*b^2, nor as a^2+3*b^2, nor as a^2+7*b^2, with a > 0 and b > 0.
+10
3
1, 14, 15, 30, 35, 42, 46, 47, 55, 60, 62, 69, 70, 78, 87, 94, 95, 105, 110, 115, 119, 120, 126, 135, 138, 140, 141, 142, 143, 154, 155, 158, 159, 165, 167, 168, 174, 182, 186, 188, 190, 195, 206, 210, 213, 215, 220, 222, 230, 231, 235, 238, 240, 248, 254, 255, 266, 270, 276, 280, 282, 285, 286, 287, 295, 299
COMMENTS
If a composite number C, in case, can be written in the form C = a^2+k*b^2, for some integers a & b, then every prime factor P (for C) being raised to an odd power can be written in the form P = c^2+k*d^2, for some integers c & d.
This statement is only true for k = 1, 2, 3.
For k = 7, with the exception of the prime factor 2, the statement mentioned above is true.
Perfect squares which can be written in all the four forms a^2+b^2, a^2+2*b^2, a^2+3*b^2 and a^2+7*b^2, with a > 0 and b > 0.
+10
3
3600, 4624, 12100, 12321, 14400, 18496, 20449, 24336, 26896, 30276, 32400, 37249, 41616, 46225, 48400, 49284, 51076, 57600, 73984, 75076, 81796, 85264, 90000, 97344, 101124, 106929, 107584, 108900, 110889, 112225, 113569, 115600, 121104, 126736, 129600, 139876, 144400, 148225, 148996, 150544, 165649, 166464, 176400, 184041, 184900, 193600, 197136
COMMENTS
If a composite number C, say, can be written in the form C = a^2+k*b^2, for some integers a & b, then every prime factor P (for C) being raised to an odd power can be written in the form P = c^2+k*d^2, for some integers c & d.
This statement is only true for k = 1, 2, 3.
For k = 7, with the exception of the prime factor 2, the statement mentioned above is true.
Numbers whose squares can be written in all the four forms a^2 + b^2, a^2 + 2*b^2, a^2 + 3*b^2 and a^2 + 7*b^2, with a > 0 and b > 0.
+10
1
60, 68, 110, 111, 120, 136, 143, 156, 164, 174, 180, 193, 204, 215, 220, 222, 226, 240, 272, 274, 286, 292, 300, 312, 318, 327, 328, 330, 333, 335, 337, 340, 348, 356, 360, 374, 380, 385, 386, 388, 407, 408, 420, 429, 430, 440, 444, 452, 457, 466, 468, 476, 480, 492, 522, 540, 544, 548, 550, 551, 555, 559, 562, 572, 579, 584
COMMENTS
If a composite number C can be written in the form C = a^2 + k*b^2, for some integers a and b, then every prime factor P (for C) being raised to an odd power can be written in the form P = c^2 + k*d^2, for some integers c and d.
This statement is only true for k = 1, 2, 3.
For k = 7, with the exception of the prime factor 2, the statement mentioned above is true.
MAPLE
filter:= proc(n) local L, x, y;
select(t -> subs(t, x*y) > 0, [isolve(n^2=x^2+y^2)]) <> []
and select(t -> subs(t, x*y) > 0, [isolve(n^2=x^2+2*y^2)]) <> []
and select(t -> subs(t, x*y) > 0, [isolve(n^2=x^2+3*y^2)]) <> []
and select(t -> subs(t, x*y) > 0, [isolve(n^2=x^2+7*y^2)]) <> []
end proc:
MATHEMATICA
okQ[n_] := Module[{x, y}, AllTrue[{1, 2, 3, 7}, Solve[x > 0 && y > 0 && n^2 == x^2 + #*y^2, {x, y}, Integers] =!= {}&]];
Perfect squares which can be written neither as a^2+b^2, nor as a^2+2*b^2, nor as a^2+3*b^2, nor as a^2+7*b^2, with a > 0 and b > 0.
+10
0
1, 2209, 27889, 96721, 146689, 229441, 253009, 418609, 516961, 703921, 786769, 966289, 1324801, 1495729, 1739761, 2211169, 2283121, 2430481, 3323329, 3411409, 4255969, 4879681, 5527201, 5755201, 7091569, 7219969, 8427409, 8994001, 9138529, 10029889, 10182481, 11282881, 11607649, 12439729, 13476241, 14922769, 15295921
COMMENTS
If a composite number C, in case, can be written in the form C = a^2+k*b^2, for some integers a & b, then every prime factor P (for C) being raised to an odd power can be written in the form P = c^2+k*d^2, for some integers c & d.
This statement is only true for k = 1, 2, 3.
For k = 7, with the exception of the prime factor 2, the statement mentioned above is true.
Numbers whose squares can be written neither as a^2 + b^2, nor as a^2 + 2*b^2, nor as a^2 + 3*b^2, nor as a^2 + 7*b^2, with a > 0 and b > 0.
+10
0
1, 47, 167, 311, 383, 479, 503, 647, 719, 839, 887, 983, 1151, 1223, 1319, 1487, 1511, 1559, 1823, 1847, 2063, 2209, 2351, 2399, 2663, 2687, 2903, 2999, 3023, 3167, 3191, 3359, 3407, 3527, 3671, 3863, 3911, 4007, 4079, 4583, 4679, 4703, 4751, 4871, 4919, 5039, 5087, 5351, 5519, 5591, 5711, 5879, 5927
COMMENTS
If a composite number C can be written in the form C = a^2+k*b^2, for some integers a and b, then every prime factor P (for C) being raised to an odd power can be written in the form P = c^2+k*d^2, for some integers c and d.
This statement is only true for k = 1, 2, 3.
For k = 7, with the exception of the prime factor 2, the statement mentioned above is true.
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