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Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.
erdos_699_finite_prime_pairs.json
json
{
"scope": "Exact finite exponent-pair certificate; no binomial rows enumerated.",
"records": [
{
"pair": [
17,
19
],
"height_cap_power_of_ten": 976,
"maximum_difference": 24,
"reciprocal_exponent_denominator": 3,
"height_threshold": 4333,
"p_power_count": 793,
"q_power_count": 763,
"exponent_pairs_considered": 603504,
"exponent_pairs_after_safe_filter": 202191,
"signed_differences_considered": 9705168,
"exact_candidate_tests": 3,
"exception_count": 3,
"exceptions": [
{
"p_exponent": 2,
"q_exponent": 2,
"signed_difference": 1,
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"q_cofactor": 4,
"smaller_height": 1444,
"larger_height": 1445
},
{
"p_exponent": 2,
"q_exponent": 2,
"signed_difference": 2,
"p_cofactor": 10,
"q_cofactor": 8,
"smaller_height": 2888,
"larger_height": 2890
},
{
"p_exponent": 2,
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"signed_difference": 3,
"p_cofactor": 15,
"q_cofactor": 12,
"smaller_height": 4332,
"larger_height": 4335
}
],
"conclusion": "max(u,v)^3 >= min(p^a*u,q^b*v) throughout the stated height range"
},
{
"pair": [
2,
17
],
"height_cap_power_of_ten": 976,
"maximum_difference": 24,
"reciprocal_exponent_denominator": 3,
"height_threshold": 49131,
"p_power_count": 3242,
"q_power_count": 793,
"exponent_pairs_considered": 2564496,
"exponent_pairs_after_safe_filter": 859352,
"signed_differences_considered": 41248896,
"exact_candidate_tests": 11,
"exception_count": 11,
"exceptions": [
{
"p_exponent": 7,
"q_exponent": 2,
"signed_difference": -4,
"p_cofactor": 9,
"q_cofactor": 4,
"smaller_height": 1152,
"larger_height": 1156
},
{
"p_exponent": 8,
"q_exponent": 2,
"signed_difference": -8,
"p_cofactor": 9,
"q_cofactor": 8,
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"larger_height": 2312
},
{
"p_exponent": 9,
"q_exponent": 2,
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"q_cofactor": 16,
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"larger_height": 4624
},
{
"p_exponent": 10,
"q_exponent": 3,
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"q_cofactor": 5,
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},
{
"p_exponent": 11,
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},
{
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"q_cofactor": 10,
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"larger_height": 49152
},
{
"p_exponent": 12,
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"smaller_height": 24565,
"larger_height": 24576
},
{
"p_exponent": 12,
"q_exponent": 3,
"signed_difference": 22,
"p_cofactor": 12,
"q_cofactor": 10,
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"larger_height": 49152
},
{
"p_exponent": 13,
"q_exponent": 3,
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"q_cofactor": 5,
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"larger_height": 24576
},
{
"p_exponent": 13,
"q_exponent": 3,
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"q_cofactor": 10,
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},
{
"p_exponent": 14,
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"signed_difference": 22,
"p_cofactor": 3,
"q_cofactor": 10,
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"larger_height": 49152
}
],
"conclusion": "max(u,v)^3 >= min(p^a*u,q^b*v) throughout the stated height range"
},
{
"pair": [
11,
23
],
"height_cap_power_of_ten": 976,
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"exponent_pairs_after_safe_filter": 224179,
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"exact_candidate_tests": 1,
"exception_count": 1,
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{
"p_exponent": 3,
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"q_cofactor": 5,
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"larger_height": 2662
}
],
"conclusion": "max(u,v)^3 >= min(p^a*u,q^b*v) throughout the stated height range"
},
{
"pair": [
5,
17
],
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{
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},
{
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{
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}
],
"conclusion": "max(u,v)^3 >= min(p^a*u,q^b*v) throughout the stated height range"
},
{
"pair": [
7,
17
],
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{
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},
{
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},
{
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{
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}
],
"conclusion": "max(u,v)^3 >= min(p^a*u,q^b*v) throughout the stated height range"
},
{
"pair": [
2,
3
],
"height_cap_power_of_ten": 976,
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"exception_count": 30,
"exceptions": [
{
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"larger_height": 32
},
{
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},
{
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{
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},
{
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{
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{
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},
{
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{
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{
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{
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{
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{
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{
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{
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{
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{
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{
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{
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{
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{
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{
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{
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{
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{
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{
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{
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{
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{
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{
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}
],
"conclusion": "max(u,v)^3 >= min(p^a*u,q^b*v) throughout the stated height range"
},
{
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2,
5
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{
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{
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{
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{
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{
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{
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{
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{
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{
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{
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{
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{
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{
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{
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{
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{
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{
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{
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{
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{
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{
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{
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{
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{
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{
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{
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{
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{
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{
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{
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}
],
"conclusion": "max(u,v)^3 >= min(p^a*u,q^b*v) throughout the stated height range"
},
{
"pair": [
3,
5
],
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{
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{
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{
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{
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{
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{
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{
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{
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{
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],
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},
{
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{
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{
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],
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},
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]
}
Graph